text:
\section{Introduction}\label{sec:In}

This paper is a survey on the adaptive
non parametric estimation
methods
for the general semi-martingale regression model in continuous time
defined as
\begin{equation}\label{sec:In.1}
 \d\,y_\zs{t} = S(t)\d\,t + \,
\d \xi_\zs{t}\,,\quad
 0\le t \le n\,,
\end{equation}
where $S(\cdot)$ is an unknown $1$ - periodic
 function,
$(\xi_\zs{t})_\zs{0\le t\le n}$  is an
unobservable noise defined by  semimartingale  with the values in the
Skorokhod space $\cD[0,n]$
 such that, for any
function $f$ from $\cL_\zs{2}[0,n]$, the stochastic integral
\begin{equation*}\label{sec:In.2}
I_\zs{n}(f)=\int^n_\zs{0}f(s)\d \xi_\zs{s}
 \end{equation*} is
well defined and has the following properties
\begin{equation}\label{sec:In.3}
\E_\zs{Q} I_\zs{n}(f)=0 \quad\mbox{and}\quad \E_\zs{Q}
I^2_\zs{n}(f)\le \kappa_\zs{Q} \int^{n}_\zs{0}\,f^{2}(s)\d s
\,.
\end{equation}
 We use $\E_\zs{Q}$ for the expectation with respect to the distribution
 $Q$ in $\cD[0,n]$ of the process $(\xi_\zs{t})_\zs{0\le t\le n}$,
 which is assumed to belong to some
 probability family $\cQ_\zs{n}$ and $\kappa_\zs{Q}$ is
some positive constant depending on the distribution $Q$.
The problem consists to estimate the function $S$ on the observations $(y_\zs{t})_\zs{0\le t\le n}$.
Note that if $(\xi_\zs{t})_\zs{0\le t\le n}$ is a  brownian motion, then we obtain the well known "signal+white noise" model
which is very popular in statistical radio-physics (see, for example, \cite{IbragimovKhasminskii1981, Kutoyants1977, Kutoyants1984, Pinsker1981}).
 In this paper we assume that in addition
 to the intrinsic noise in the radio-electronic system, approximated usually by the Gaussian white or color noise,
 the useful signal $S$ is distorted by the impulse flow described by the processes with jumps.
The cause of the appearance of a pulse stream in the radio-electronic systems can be, for example, either external unintended (atmospheric) or intentional impulse noise
 and
the errors in the demodulation and the channel decoding for the binary information symbols. Note that,
for the first time the impulse noises
for the detection signal problems
 have been introduced
 on the basis of the compound Poisson processes
was introduced  by Kassam in
\cite{Kassam1988}.
 Later, such processes was used
  in
  \cite{KonevPergamenshchikov2012, KonevPergamenshchikov2015, KPP2014, Pchelintsev2013}
for the
parametric and
nonparametric signal estimation problems.
 However,  the compound Poisson process can  describe only   the large impulses  influence of fixed  single frequency.
 Taking into account  that in the telecommunication systems,  the impulses are without limitations on frequencies
 one needs to extend the framework of the observation model by making use the  L\'evy
 processes  \eqref{sec:In.1+1} which is a particular case of the general semimartinagale regression model
 introduced in
\cite{KonevPergamenshchikov2009a}.
Generally, we
 consider  nonparametric estimation problems for the function $S$ from
$\cL_\zs{2}$ under the condition that
the distribution of the noise $(\xi_\zs{t})_\zs{0\le t\le n}$ is unknown. We know only that this distribution belongs to
 some distribution family $\cQ_\zs{n}$. In this case
we use the robust estimation approach
proposed  in \cite{GaltchoukPergamenshchikov2006, KonevPergamenshchikov2012, KonevPergamenshchikov2015}
for the nonparametric estimation. According to this approach we have to construct an estimator $\wh{S}_\zs{n}$
 (any function of $(y_\zs{t})_\zs{0\le t\le n}$) for $S$ to minimize the robust risk defined as
\begin{equation}\label{sec:risks}
\cR^{*}_\zs{n}(\wh{S}_\zs{n},S)=\sup_\zs{Q\in\cQ_\zs{n}}\,
\cR_\zs{Q}(\wh{S}_\zs{n},S)\,,
\end{equation}
where  $\cR_\zs{Q}(\cdot,\cdot)$ is the usual quadratic risk of the form
\begin{equation}\label{sec:risks_00}
\cR_\zs{Q}(\wh{S}_\zs{n},S):=
\E_\zs{Q}\,\|\wh{S}_\zs{n}-S\|^2
\quad\mbox{and}\quad
\Vert S\Vert^{2}=\int^{1}_\zs{0}\,S^{2}(t)\d t
\,.
\end{equation}
It is clear that if we don't know the distribution
of the observation one  needs to find an estimator which will be optimal for all possible observation distributions.
Moreover in this paper we consider the estimation problem in the adaptive setting,
i.e. when the regularity of $S$ is unknown.
To this end
 we use the adaptive method based on the model selection approach.
The interest to such statistical procedures is explained by the fact that
they provide
adaptive solutions for the nonparametric estimation through
oracle inequalities which give the  non-asymptotic upper
bound for the quadratic risk including
the minimal risk over chosen  family of estimators.
It should be noted that for the first time the model selection methods
were proposed by
 Akaike \cite{Akaike1974} and Mallows \cite{Mallows1973}
for parametric models. Then,
these methods had been developed
for the nonparametric estimation
 and
the oracle inequalities for the quadratic risks was obtained  by
Barron, Birg\'e and Massart \cite{BarronBirgeMassart1999},
Massart \cite{Massart2005},
by Fourdrinier and Pergamenshchikov \cite{FourdrinierPergamenshchikov2007}
for the regression models in discrete time
and
\cite{KonevPergamenshchikov2010} in continuous time.
Unfortunately, the oracle inequalities obtained in these papers
 can not
provide the efficient estimation in the adaptive setting, since
the upper bounds in these inequalities
have some fixed coefficients in the main terms which are more than one.
To obtain  the efficiency property for estimation
procedures  one has to obtain
 the sharp oracle inequalities, i.e.
 in which
  the factor at the principal term on the right-hand side of the inequality
 is close to unity.
The first result on sharp inequalities is most likely due to Kneip \cite{Kneip1994}
who studied a Gaussian regression model in the discrete time.
It will be observed that the
derivation of oracle inequalities usually rests upon
the fact that the initial model, by applying the Fourier transformation,
can be reduced to the Gaussian independent observations.
However, such  transformation is possible only for
Gaussian models with independent homogeneous observations or for
 inhomogeneous ones with  known correlation characteristics.
For the general non Gaussian observations one needs to use the approach proposed by
 Galtchouk and Pergamenshchikov
 \cite{GaltchoukPergamenshchikov2009a, GaltchoukPergamenshchikov2009b}  for the heteroscedastic
 regression models in discrete time and developed then by
Konev and Pergamenshchikov in
\cite{KonevPergamenshchikov2009a, KonevPergamenshchikov2009b, KonevPergamenshchikov2012, KonevPergamenshchikov2015}
for semimartingale models in  continuous time.
In general the model selection is an
adaptive rule $\wh{\alpha}$ which choses an estimator $S^{*}=\wh{S}_\zs{\wh{\alpha}}$ from
an estimate family
$(\wh{S}_\zs{\alpha})_\zs{\alpha\in\cA}$. The goal of  this selection is to prove the following
nonasymptotic
oracle inequality: for any sufficient small $\delta >0$ and any observation duration $n\ge 1$
\begin{equation}\label{sec:In.3+5}
\cR( S^{*},S)\,
\le\,(1+\delta)\,\min_\zs{\alpha\in\cA}\,\cR(\wh{S}_\zs{\alpha},S)+\delta^{-1}
\cB_\zs{n}\,,
\end{equation}
where the rest term
$\cB_\zs{n}$
 is sufficiently small with respect to the minimax convergence rate.
Such oracle inequalities are called {\sl sharp}, since the coefficient in the main term
$1+\delta$
is close to one for sufficiently small $\delta>0$.
Moreover, in this paper we represent the new results on the improved estimation methods for the
nonparametric models \eqref{sec:In.1}.
Usually, the model selection procedures are based on the least squares estimators.  But in
\cite{PchelintsevPchelintsev_VPergamenshchikov2017}
 it is propose
 to use the improved least square estimators which enable  to improve considerably the non asymptotic estimation accuracy.
At the first time such idea  was proposed in \cite{FourdrinierPergamenshchikov2007}
for the regression in discrete time and in
\cite{KonevPergamenshchikov2010}
for the Gaussian regression model in continuous time. In
\cite{PchelintsevPchelintsev_VPergamenshchikov2017}
 these methods
are developed
for the non - Gaussion
regression models
in continuous time.
It should be noted that generally for the conditionally Gaussian regression models  one can not use the well known improved estimators proposed in
\cite{JamesStein1961, FourdrinierStrawderman1996} for Gaussian or spherically symmetric observations. To apply the improved estimation methods to the
 non Gaussian regression models in continuous time
 one needs to modify the well known James--Stein procedure in the way  proposed  in  \cite{Pchelintsev2013, KPP2014}.
 For the improved model selection procedures the oracle inequality \eqref{sec:In.3+5}
  is shown also. We note that this
inequality allows us to provide the asymptotic efficiency without knowing the regularity of the function being estimated.
The efficacy property for a nonparametric estimate $S^{*}$ means
$$
\lim_\zs{n\to\infty}\,
\upsilon_\zs{n}\,\sup_\zs{S\in W^k_\zs{r}}\,
\cR(S^{*},S)\,
=
\lim_\zs{n\to\infty}\,
\upsilon_\zs{n}
\,
\inf_\zs{\wh{S}}
\sup_\zs{S\in W^k_\zs{r}}\,
\cR(\wh{S},S)\,
=
l_\zs{*}(\r)\,,
$$
where
\begin{equation}\label{sec:Ae.3}
l_\zs{*}(\r)\,=\,((1+2k)\r)^{1/(2k+1)}\,
\left(\frac{k}{\pi (k+1)}\right)^{2k/(2k+1)}
\,,
\end{equation}
$\upsilon_\zs{n}$ is a normalizing coefficient (convergence rate),
 $W^{k}_\zs{r}$ is the Sobolev ball of a radius  $r>0$ and the regularity  $k\ge 1$. The limit \eqref{sec:Ae.3}
 is called the Pinsker constant
 which is calculated by Pinsker in \cite{Pinsker1981}.

The rest of the paper is organized as follows. In the next section \ref{Examples}, we describe
the L\'evy, Ornstein--Uhlenbeck and semi-Markov processes as the examples of a
semimartingale impusle noise in the model \eqref{sec:In.1}. In Section \ref{sec:Mo} we construct the model selection
procedure based on the least square estimators and show the sharp oracle inequalities.
In Section  \ref{sec:Imp} we give the improved least squares estimators and
we study the improvement effect for the semimartingale model \eqref{sec:In.1}. In Section \ref{sec:IMo} we construct the improved model selection
procedure and show the sharp oracle inequalities. The asymptotic efficiency is studed in Section \ref{sec:Ae}.

\section{Examples}\label{Examples}

\subsection{L\'evy model}

First we consider the model \eqref{sec:In.1} with the L\'evy noise process, i.e. we assume that
 the noise process $(\xi_\zs{t})_\zs{0\le t\le n}$
is defined as
\begin{equation}\label{sec:In.1+1}
\xi_\zs{t}=\varrho_\zs{1} w_\zs{t} + \varrho_\zs{2} z_\zs{t}
\quad\mbox{and}\quad
z_\zs{t}=x*(\mu-\wt{\mu})_\zs{t}
\,,
\end{equation}
where $\varrho_\zs{1}$ and $\varrho_\zs{2}$ are  some unknown constants,
$(w_\zs{t})_\zs{t\ge\,0}$ is a standard brownian motion,
 $\mu(\d s\,\d x)$ is a jump measure with  deterministic
compensator $\wt{\mu}(\d s\,\d x)=\d s\Pi(\d x)$,
$\Pi(\cdot)$ is a L\'evy measure, i.e.  some positive measure on $\bbr_\zs{*}=\bbr\setminus \{0\}$,
(see, for example,
\cite{JacodShiryaev2002, ContTankov2004} for details) such that
\begin{equation*}\label{sec:Ex.1-00_mPi}
\Pi(x^{2})=1
\quad\mbox{and}\quad
\Pi(x^{6})
\,<\,\infty\,.
\end{equation*}
Here we use the notation $\Pi(\vert x\vert^{m})=\int_\zs{\bbr_\zs{*}}\,\vert z\vert^{m}\,\Pi(\d z)$. Note that the L\'evy measure
 $\Pi(\bbr_\zs{*})$ could be equal to $+\infty$.
 One can check directly that for the process \eqref{sec:In.1+1}
the condition \eqref{sec:In.3} holds with  $\kappa_\zs{Q}=\sigma_\zs{Q}=\varrho^{2}_\zs{1}+\varrho^{2}_\zs{2}$.
We assume that the nuisance parameters
    $\varrho_\zs{1}$
and $\varrho_\zs{2}$ of the process $(\xi_\zs{t})_\zs{0\le t\le n}$
 satisfy the conditions
\begin{equation}\label{sec:Ex.01-1}
0< \underline{\varrho}\le \varrho^{2}_\zs{1}
\quad\mbox{and}\quad
\sigma_\zs{Q}\,
\le
\varsigma^{*}
\,,
\end{equation}
where
 the bounds
 $\underline{\varrho}$
and
$\varsigma^{*}$
are functions of $n$, i.e.
 $\underline{\varrho}= \underline{\varrho}_\zs{n}$
and
$\varsigma^{*}=\varsigma^{*}_\zs{n}$
 such that for any $\check{\delta}>0$
\begin{equation}\label{sec:Ex.01-2}
\liminf_\zs{n\to\infty}\,n^{\check{\delta}}\,
\underline{\varrho}_\zs{n}
\,
>0
\quad\mbox{and}\quad
\lim_\zs{n\to\infty}\,n^{-\check{\delta}}\,\varsigma^{*}_\zs{n}
=0
\,.
\end{equation}

For this example $\cQ_\zs{n}$ is  the family
of all distributions of process \eqref{sec:In.1} -- \eqref{sec:In.1+1} on the Skorokhod space
$\D[0,n]$ satisfying the conditions \eqref{sec:Ex.01-1} -- \eqref{sec:Ex.01-2}.

The models \eqref{sec:In.1}
with the L\'evy's type noise
are used in different applied problems
(see \cite{Bertoin1996}, for details). Such  models naturally arise
in the nonparametric functional statistics problems
(see, for example, \cite{FerratyVieu2006}).

\subsection{Ornstein -- Uhlenbeck model}

Now we consider the noise process $(\xi_\zs{t})_\zs{t\ge 0}$ in
\eqref{sec:In.1} difened by a non-Gaussian Ornstein-Uhlenbeck
process with the L\'evy subordinator. Let the noise
process in \eqref{sec:In.1} obey the equation
\begin{equation}\label{sec:Ex.1}
\d\xi_\zs{t} = a\xi_\zs{t}\d t+\d u_\zs{t}\,,\quad \xi_\zs{0}=0\,,
\end{equation}
where
$
u_\zs{t}=
\varrho_\zs{1}\, w_\zs{t}+\varrho_\zs{2}\,z_\zs{t}$ and
the process
$z_\zs{t}$
is defied in \eqref{sec:In.1+1}.
Here $a\le 0$,
$\varrho_\zs{1}$ and $\varrho_\zs{2}$ are unknown parameters.
 We assume that the parameters  $\varrho_\zs{1}$
and $\varrho_\zs{2}$ satisfy the conditions
\eqref{sec:Ex.01-1} and the parameter
\begin{equation}\label{sec:Ex.3+}
-a_\zs{max}\le  a\le 0\,,
\end{equation}
where the bound $a_\zs{max}>0$ is the function of $n$, i.e. $a_\zs{max}=a_\zs{max}(n)$,
such that for any $\check{\delta}>0$
\begin{equation}\label{sec:Ex.3-1}
\lim_\zs{n\to\infty}\,\frac{a_\zs{max}}{n^{\check{\delta}}}=0
\,.
\end{equation}
In this case  $\cQ_\zs{n}$ is the family
of all distributions of process  \eqref{sec:Ex.1} on the Skorokhod space
$\D[0,n]$ satisfying the conditions \eqref{sec:Ex.01-1},  \eqref{sec:Ex.3+} and \eqref{sec:Ex.3-1}.
Note also that the processes
\eqref{sec:In.1+1} and
\eqref{sec:Ex.1} are $\cG$ - conditionally  Gaussian square integrated semimartingales,  where
 $\cG=\sigma\{z_\zs{t}\,,\,t\ge 0\}$.

Such processes are used in
the financial Black-Scholes type markets with  jumps (see, for
example, \cite{BarndorffNielsenShephard2001, DelongKluppelberg2008} and the references therein).
Note also that in the case when $\varrho_\zs{2}=0$ for the parametric estimation problem such models are considered in
\cite{HopfnerKutoyants2009, HopfnerKutoyants2010, KonevPergamenshchikov2003}.

\subsection{Semi -- Markov model}

In
\cite{BarbuBeltaifPergamenshchikov2017a, BarbuBeltaifPergamenshchikov2017b}
it is introduced
the regression model   \eqref{sec:In.1}  in which
 the noise process describes by the equation
\begin{equation}\label{sec:Ex.5}
 \xi_\zs{t} = \varrho_\zs{1} L_\zs{t} + \varrho_\zs{2} X_\zs{t}\,,
\end{equation}
and  the L\'evy  process $L_\zs{t}$ is defined as
 \begin{equation*}\label{sec:Mcs.1}
L_\zs{t}=\check{\varrho}\,w_\zs{t}
+
\sqrt{1-\check{\varrho}^{2}}\,z_\zs{t}\,,
\end{equation*}
 where $0\le \check{\varrho}\le 1$ is an unknown constant.
Moreover, we assume that the pure jump process $(X_\zs{t})_\zs{t\ge\, 0}$ in
 \eqref{sec:Ex.5}
 is a semi-Markov process with the following form
 \begin{equation}\label{sec:Ex.2}
 X_\zs{t} = \sum_\zs{i=1}^{N_\zs{t}} Y_\zs{i},
\end{equation}
where $(Y_\zs{i})_\zs{i\ge\, 1}$ is an i.i.d. sequence of random variables with
\begin{equation*}\label{sec:Ex.3+?++}
\E\,Y_\zs{i}=0\,,\quad
\E\,Y^2_\zs{i}=1
\quad\mbox{and}\quad
\E\,Y^4_\zs{i}<\infty
\,.
\end{equation*}
Here $N_\zs{t}$ is a general  counting process (see, for example, \cite{Mikosch2004})
defined as
\begin{equation*}\label{sec:Ex.4}
N_\zs{t} = \sum_\zs{k=1}^{\infty} \Chi_\zs{\{T_\zs{k} \le t\}}
\quad\mbox{and}\quad
T_\zs{k}=\sum_\zs{l=1}^k\, \tau_\zs{l}\,,
\end{equation*}
where $(\tau_\zs{l})_\zs{l\ge\,1}$ is an i.i.d. sequence of positive integrated
 random variables with distribution $\eta$ and mean $\check{\tau}=\E\,\tau_\zs{1}>0$. We assume that the processes
$(N_\zs{t})_\zs{t\ge 0}$ and  $(Y_\zs{i})_{i\ge\, 1}$ are independent between them and are also independent of $(L_\zs{t})_\zs{t\ge 0}$.
Here, the family  $\cQ_\zs{n}$ is defined by
of all distributions of process  \eqref{sec:Ex.5} on the Skorokhod space
$\D[0,n]$ with the parameters $\varrho_\zs{1}$ and
$\varrho_\zs{2}$
 satisfying the conditions \eqref{sec:Ex.01-1}  and $0\le \check{\varrho}\le 1$.

Note that the process $(X_\zs{t})_\zs{t\ge\, 0}$ is a special case of a semi-Markov process (see, e.g., \cite{BarbuLimnios2008} and \cite{LimniosOprisan}).
 It should be noted that if $\tau_\zs{j}$ are exponential random variables, then $(N_\zs{t})_\zs{t\ge 0}$
is a Poisson process and, in this case, $(\xi_\zs{t})_\zs{t\ge 0}$ is a L\'evy process.
But, in the general case when the process \eqref{sec:Ex.2} is not a L\'evy process, this process has a memory and cannot be treated
in the framework of semi-martingales with independent increments.  In this case, we need to
develop new tools based on renewal theory arguments from  \cite{Goldie1991}.

It should be noted  that for $\check{\varrho}>0$  the process \eqref{sec:Ex.5}
 is $\cG$ - conditionally  Gaussian also. In this case
 $\cG=\sigma\{z_\zs{t}\,,\,X_\zs{t}\,,\,t\ge 0\}$.

\section{Model selection}\label{sec:Mo}

\noindent
Let $(\phi_\zs{j})_\zs{j\ge\, 1}$ be an orthonormal uniformly bounded basis in $\cL_\zs{2}[0,1]$, i.e.
for some  constant $\phi_\zs{*}\ge 1$, which may be depend on $n$,
\begin{equation*}\label{sec:In.3-00}
\sup_\zs{0\le j\le n}\,\sup_\zs{0\le t\le 1}\vert\phi_\zs{j}(t)\vert\,
\le\,
\phi_\zs{*}
<\infty\,.
\end{equation*}
For example, we can take
 the trigonometric basis    defined as $\Tr_\zs{1}\equiv 1$ and, for $j\ge 2,$
\begin{equation}\label{sec:In.5}
 \Tr_\zs{j}(x)= \sqrt 2
\left\{
\begin{array}{c}
\cos(2\pi[j/2] x)\, \quad\mbox{for even}\quad j;\\[4mm]
\sin(2\pi[j/2] x)\quad\mbox{for odd}\quad j,
\end{array}
\right.
\end{equation}
where $[x]$ denotes the integer part of $x$.

To estimate the function $S$ we use here the model selection procedure for continuous time regression models from
\cite{KonevPergamenshchikov2012} based on the Fourrier expansion. We recall that
for any function $S$ from $\cL_\zs{2}[0,1]$ we can write
\begin{equation*}\label{sec:In.5+++Fourrier}
S(t)=\sum^{\infty}_\zs{j=1}\,\theta_\zs{j}\,\phi_\zs{j}(t)
\quad\mbox{and}\quad
\theta_\zs{j}= (S,\phi_\zs{j}) = \int_\zs{0}^{1} S(t) \phi_\zs{j}(t)\d t
\,.
\end{equation*}
So, to estimate the function $S$ it suffices to estimate the coefficients $\theta_\zs{j}$ and to replace them in this representation by their estimators.
Using the fact that the function $S$ and $\phi_\zs{j}$ are $1$ - periodic we can write that
$$
\theta_\zs{j}=\frac{1}{n} \int_\zs{0}^{n}\, \phi_\zs{j}(t)\,S(t) \d t
\,.
$$
If we replace here the differential $S(t)\d t$ by the stochastic observed differential $\d y_\zs{t}$
then we obtain the natural estimate for $\theta_\zs{j}$ on the time interval $[0,n]$
\begin{equation*}\label{sec:In.7}
\wh{\theta}_\zs{j,n}= \frac{1}{n} \int_\zs{0}^{n}  \phi_\zs{j}(t) \d\,y_\zs{t}\,,
\end{equation*}
which
can be represented, in view of the model \eqref{sec:In.1}, as
\begin{equation*}\label{sec:In.8}
\wh{\theta}_\zs{j,n}= \theta_\zs{j} + \frac{1}{\sqrt n}\xi_\zs{j,n}
\quad\mbox{and}\quad
\xi_\zs{j,n}
= \frac{1}{\sqrt n} I_\zs{n}(\phi_\zs{j})\,.
\end{equation*}
We need to impose some stability conditions for the noise Fourier transform sequence
$(\xi_\zs{j,n})_\zs{1\le j\le n}$. To this end we set  for some stability noise intensity parameter $\sigma_\zs{Q}> 0$
the following function
\begin{equation}
\label{L_1_Q}
\L_\zs{1,n}(Q)=
 \sup_\zs{x\in [-1,1]^{n}}\,
 \left|
\sum^{n}_\zs{j=1}\,x_\zs{j}\,
\left(
\E_\zs{Q}\,\xi^{2}_\zs{j,n}
-
\sigma_\zs{Q}
\right)
\right|
\,.
\end{equation}
In \cite{KonevPergamenshchikov2012}
the  parameter $\sigma_\zs{Q}$ is called proxy variance .

\noindent $\C_\zs{1})$ {\it There exists a variance proxy
$\sigma_\zs{Q}> 0$
such that for any $\epsilon>0$
$$
\lim_\zs{n\to\infty}\frac{\L_\zs{1,n}(Q)}{n^{\epsilon}}
=0\,.
$$
}

\noindent Moreover, we set
\begin{equation*}
\label{L_2_Q}
\L_\zs{2,n}(Q)=
 \sup_\zs{|x|\le 1}
\E_\zs{Q}\, \left(
\sum^{n}_\zs{j=1}\,x_\zs{j}\,
(\xi^2_\zs{j,n}-\E_\zs{Q} \xi^2_\zs{j,n})
\right)^2\,.
\end{equation*}
\noindent $\C_\zs{2})$ {\it Assume that for any $\epsilon>0$
$$
\lim_\zs{n\to\infty}\frac{\L_\zs{2,n}(Q)}{n^{\epsilon}}
=0\,.
$$
}

Now (see, for example, \cite{IbragimovKhasminskii1981}) we can estimate the function $S$ by the projection estimators, i.e.
\begin{equation}\label{sec:In.7++pre}
\wh{S}_\zs{m}(t)=\sum^{m}_\zs{j=1}\,\wh{\theta}_\zs{j,n}\,\phi_\zs{j}(t)\,,\quad 0\le t\le 1\,,
\end{equation}
for some number $m\to\infty$ as $n\to\infty$. It should be noted that Pinsker in \cite{Pinsker1981} shows that the projection estimators of the form
\eqref{sec:In.7++pre}
 are not efficient. For obtaining efficient estimation one needs to use weighted least square estimators defined as
\begin{equation}\label{sec:Mo.1+??+}
\wh{S}_\lambda (t) = \sum_\zs{j=1}^{n} \lambda(j) \wh{\theta}_\zs{j,n} \phi_\zs{j}(t)\,,
\end{equation}
where the coefficients $\lambda=(\lambda(j))_\zs{1\le j\le n}$ belong to some finite set $\Lambda$ from $[0,1]^n$.
 As it is shown in  \cite{Pinsker1981}, in order to obtain efficient estimators, the coefficients $\lambda(j)$ in \eqref{sec:Mo.1+??+} need to be chosen depending on
the regularity of the unknown function $S$. Since we consider the adaptive case, i.e. we assume that the regularity of the function $S$ is unknown, then we chose the weight coefficients on the basis of  the model selection procedure proposed in \cite{KonevPergamenshchikov2012}
   for the general semi-martingale regression model in continuous time.
 To the end, first we set
 \begin{equation*}\label{sec:Mo.2++???}
\nu=\#(\Lambda)
\quad\mbox{and}\quad
\vert\Lambda\vert_\zs{*}=1+ \max_\zs{\lambda\in\Lambda}\,
\sum^{n}_\zs{j=1}\lambda(j)
\,,
\end{equation*}
where $\#(\Lambda)$ is the cardinal number of  $\Lambda$.
Now, to choose a weight sequence $\lambda$ in the set $\Lambda$ we use the empirical  quadratic risk, defined as
$$
\Er_n(\lambda) = \parallel \wh{S}_\lambda-S\parallel^2,
$$
which in our case is equal to
\begin{equation*}\label{sec:Mo.3}
\Er_n(\lambda) = \sum_\zs{j=1}^{n} \lambda^2(j) \wh{\theta}^2_\zs{j,n} -2 \sum_\zs{j=1}^{n} \lambda(j) \wh{\theta}_\zs{j,n}\theta_\zs{j}+ \sum_\zs{j=1}^{\infty} \theta^2_\zs{j}.
\end{equation*}
Since the Fourier coefficients $(\theta_\zs{j})_\zs{j\ge\,1}$ are unknown, we replace
the terms $\wh{\theta}_\zs{j,n}\theta_\zs{j,n}$ by
\begin{equation*}\label{sec:Mo.4+?/}
\wt{\theta}_\zs{j,n} = \wh{\theta}^2_\zs{j,n} - \frac{  \wh{\sigma}_\zs{n}}{n}\,,
\end{equation*}
where $\wh{\sigma}_\zs{n}$ is an estimate for the variance proxy $\sigma_\zs{Q}$ defined in \eqref{L_1_Q}.
If it is known, we take $\wh{\sigma}_\zs{n}=\sigma_\zs{Q}$, otherwise, we can choose it, for example, as in \cite{KonevPergamenshchikov2012}, i.e.
\begin{equation}\label{sec:Mo.4+===}
\wh{\sigma}_\zs{n}=
\sum^n_\zs{j=[\sqrt{n}]+1}\,\wh{t}^2_\zs{j,n}\,,
\end{equation}
where $\wh{t}_\zs{j,n}$ are the estimators for the Fourier coefficients with respect to the trigonometric basis \eqref{sec:In.5}, i.e.
\begin{equation*}\label{sec:Mo.4-2-31-3}
\wh{t}_\zs{j,n}=\frac{1}{n}
\int^{n}_\zs{0}\,Tr_\zs{j}(t)\d y_\zs{t}\,.
\end{equation*}

\noindent
Finally, in order to choose the weights, we will minimize the following cost function
\begin{equation*}\label{sec:Mo.5}
J_n(\lambda)=\sum_\zs{j=1}^{n} \lambda^2(j) \wh{\theta}^2_\zs{j,n} -2 \sum_\zs{j=1}^{n} \lambda(j)\wt{\theta}_\zs{j,n} + \delta\,P_\zs{n}(\lambda),
\end{equation*}
where $\delta>0$ is some  threshold which will be specified later and the penalty term is

\begin{equation}\label{sec:Mo.6}
P_\zs{n}(\lambda)= \frac{  \wh{\sigma}_\zs{n} |\lambda|^2}{n}.
\end{equation}
\noindent
We define the model selection procedure as
\begin{equation}\label{sec:Mo.9-MS}
\wh{S}_* = \wh{S}_\zs{\hat \lambda}
\quad\mbox{with}\quad
\wh{\lambda}= \mbox{argmin}_\zs{\lambda\in\Lambda} J_n(\lambda)\,.
\end{equation}
We recall that the set $\Lambda$ is finite so $\hat \lambda$ exists. In the case when $\hat \lambda$ is not unique, we take one of them.

As is shown in
\cite{BarbuBeltaifPergamenshchikov2017a, KonevPergamenshchikov2012,  PchelintsevPchelintsev_VPergamenshchikov2017}
 both Conditions $\C_\zs{1})$
and $\C_\zs{2})$ hold for the processes \eqref{sec:In.1+1}, \eqref{sec:Ex.1} and \eqref{sec:Ex.5}.

\begin{proposition}\label{Pr.sec:2.1}
If the conditions $\C_\zs{1})$ and $\C_\zs{2})$ hold for the
distribution $Q$ of the process $\xi$ in \eqref{sec:In.1}, then, for any $n\geq1$ and $0<\delta<1/3$,
the risk \eqref{sec:risks_00} of estimate \eqref{sec:Mo.9-MS} for $S$
satisfies the oracle inequality
\begin{equation}\label{sec:Mo.15_OrIneq}
\cR_\zs{Q}(\wh{S}_\zs{*},S)\,\le\, \frac{1+3\delta}{1-3\delta}
\min_\zs{\lambda\in\Lambda} \cR_\zs{Q}(\wh{S}_\zs{\lambda},S)
+
\frac{\B_\zs{n}(Q)}{\delta n}
\,,
\end{equation}
where
$\B_\zs{n}(Q)=\U_\zs{n}(Q)
\left(
1+\vert\Lambda\vert_\zs{*}\E_\zs{Q}|\wh{\sigma}_\zs{n}-\sigma_\zs{Q}|
\right)
$
and
 the coefficient $\U_\zs{n}(Q)$ is such that for any $\epsilon>0$
\begin{equation}
\label{termB_rest}
\lim_\zs{n\to\infty}\frac{\U_\zs{n}(Q)}{n^{\epsilon}}=0
\,.
\end{equation}
\end{proposition}

\noindent
 In the case, when the value of $\sigma_\zs{Q}$ is known, one can take
$\wh{\sigma}_\zs{n}=\sigma_\zs{Q}$ and
\begin{equation*}\label{sec:Mo.9_P}
P_\zs{n}(\lambda)=\frac{\sigma_\zs{Q}\,|\lambda|^2_\zs{n}}{n}\,,
\end{equation*}
then we can rewrite the oracle inequality \eqref{sec:Mo.15_OrIneq}  with $\B_\zs{n}(Q)=\U_\zs{n}(Q)$. Also we study the accuracy properties for the estimator \eqref{sec:Mo.4+===}.

\begin{proposition}\label{sec:Mo.Prop.1}
Let in the model \eqref{sec:In.1} the function $S(\cdot)$ is continuously differentiable.
Then, for any $n\geq 2$,
\begin{equation*}\label{sec:Mo.15_kappa}
\E_\zs{Q}|\wh{\sigma}_\zs{n}-\sigma_\zs{Q}| \leq
\frac{\varkappa_\zs{n}(Q)(1+\|\dot{S}\|^2)}{\sqrt{n}}\,,
\end{equation*}
where the term $\varkappa_\zs{n}(Q)$ possesses  the property  \eqref{termB_rest}.
\end{proposition}
\noindent
To obtain the oracle inequality for the robust risk \eqref{sec:risks}
 we need some additional condition on the distribution family $\cQ_\zs{n}$.
We set
\begin{equation}
\label{sigma*_n}
\varsigma^{*}=\varsigma^{*}_\zs{n}=\sup_\zs{Q\in\cQ_\zs{n}}
\sigma_\zs{Q}
\quad\mbox{and}\quad
\L^{*}_\zs{n}=
\sup_\zs{Q\in\cQ_\zs{n}}(\L_\zs{1,n}(Q)+\L_\zs{2,n}(Q))
\,.
\end{equation}

\noindent $\C^{*}_\zs{1}$) {\it Assume that the conditions $\C_\zs{1})$--$\C_\zs{2})$
hold and for any  $\epsilon>0$
$$
\lim_\zs{n\to\infty}\frac{\L^{*}_\zs{n}+\varsigma^{*}_\zs{n}}{n^{\epsilon}}
=0\,.
$$
}

\noindent
Now we impose the conditions on the set of the weight coefficients $\Lambda$.

\noindent $\C^{*}_\zs{2})$ {\it Assume that the set $\Lambda$ is such that for any $\epsilon>0$
\begin{equation*}\label{sec:Mo.8+1}
\lim_\zs{n\to\infty}\frac{\nu}{n^{\epsilon}}=0
\quad\mbox{and}\quad
\lim_\zs{n\to\infty}\,\frac{\vert\Lambda\vert_\zs{*}}{n^{1/2+\epsilon}}
=0\,.
\end{equation*}
}

\begin{theorem}\label{Th.sec:2.3}
Assume that the conditions
$\C^*_\zs{1})$--$\C^*_\zs{2})$ hold.  Then the robust risk
 \eqref{sec:risks} of the estimate \eqref{sec:Mo.9-MS} for
continuously differentiable function $S(t)$ satisfies for any $n\ge 2$  and
$0<\delta<1/3$ the oracle inequality
\begin{align*}\label{sec:Mo.20}
\cR^{*}_\zs{n}(\wh{S}_\zs{*},S)\,\le\,
\frac{1+3\delta}{1-3\delta} \min_\zs{\lambda\in\Lambda}
\cR_n^{*}(\wh{S}_\zs{\lambda},S) +\frac{1}{\delta n}\,\B^{*}_\zs{n}(1+\|\dot{S}\|^2)\,,
\end{align*}
where the term $\B^{*}_\zs{n}$ satisfies the property
\eqref{termB_rest}.
\end{theorem}

Now we specify the weight coefficients $(\lambda(j))_\zs{j\ge
1}$ in the way proposed in \cite{GaltchoukPergamenshchikov2009a}
 for a heteroscedastic regression
model in discrete time. First we define the normalizing coefficient which defined the minimax convergence rate
\begin{equation}
\label{upsilon-nnn}
 v_\zs{n}=
\frac{n}{\varsigma^{*}}
\,,
\end{equation}
where the upper proxy variance is $\varsigma^{*}$ is defined in \eqref{sigma*_n}.  Consider a numerical grid of the form
\begin{equation*}\label{sec:Imp.7}
\cA_\zs{n}=\{1,\ldots,k^*\}\times\{r_1,\ldots,r_m\}\,,
\end{equation*}
where  $r_i=i\varepsilon$ and $m=[1/\varepsilon^2]$. Both
parameters $k^*\ge 1$ and $0<\varepsilon\le 1$ are assumed to be
functions of $n$, i.e. $ k^*=k^*(n)$ and
$\varepsilon=\varepsilon(n)$, such that for any $\delta>0$
\begin{equation*}\label{sec:Imp.8}
\left\{
\begin{array}{ll}
&\lim_\zs{n\to\infty}\,k^*(n)=+\infty\,,
\quad
\lim_\zs{n\to\infty}\,\dfrac{k^*(n)}{\ln n}=0\,,\\[6mm]
&\lim_\zs{n\to\infty}\varepsilon(n)=0
\quad\mbox{and}\quad
\lim_\zs{n\to\infty}\,n^{\delta}\varepsilon(n)\,=+\infty\, .
\end{array}
\right.
\end{equation*}
One can take, for
example,
$$
\varepsilon(n)=\frac{1}{\ln (n+1)}
\quad\mbox{and}\quad
k^*(n)=\sqrt{\ln (n+1)}\,.
$$
For each $\alpha=(\beta,r)\in\cA_\zs{n}$ we introduce the weight
sequence $\lambda_\zs{\alpha}=(\lambda_\zs{\alpha}(j))_\zs{j\ge 1}$
as
\begin{equation}\label{sec:Imp.9}
\lambda_\zs{\alpha}(j)=\Chi_\zs{\{1\le j\le d\}}+
\left(1-(j/\omega_\alpha)^\beta\right)\, \Chi_\zs{\{ d<j\le
\omega_\alpha\}}
\end{equation}
where $d=d(\alpha)=\left[\omega_\zs{\alpha}/\ln (n+1)\right]$,
$
\omega_\zs{\alpha}=\left(\tau_\zs{\beta}\,r\,v_\zs{n}\right)^{1/(2\beta+1)}$ and
$$
\tau_\zs{\beta}=\frac{(\beta+1)(2\beta+1)}{\pi^{2\beta}\beta}\,.
$$
We set
\begin{equation}\label{sec:Imp.10_Lambda}
\Lambda\,=\,\{\lambda_\zs{\alpha}\,,\,\alpha\in\cA_\zs{n}\}\,.
\end{equation}
It will be noted that in this case the cardinal of the set $\Lambda$ is
$\nu=k^{*} m$. Moreover,
taking into account that $\tau_\zs{\beta}<1$ for $\beta\ge 1$
we obtain for the set \eqref{sec:Imp.10_Lambda}
\begin{equation*}
\label{sec:Ga.1++1--2}
 \vert \Lambda\vert_\zs{*}\,
 \le\,1+
\sup_\zs{\alpha\in\cA}
  \omega_\zs{\alpha}
\le 1+(\upsilon_\zs{n}/\ve )^{1/3}\,.
\end{equation*}

Note that the form \eqref{sec:Imp.9}
for the weight coefficients
 was proposed by Pinsker in \cite{Pinsker1981}
 for the efficient estimation in the nonadaptive case, i.e. when the regularity parameters of the function $S$ are known.
In the adaptive case  these weight coefficients are
  used in \cite{KonevPergamenshchikov2012, KonevPergamenshchikov2015}
   to show the asymptotic efficiency for model selection procedures.

\section{Improved estimation}\label{sec:Imp}

In this Section we consider the  improved estimation method for the model \eqref{sec:In.1}.
We impose the following additional condition on the noise distribution.

$\D_\zs{1}$) {\sl There exists $n_\zs{0}\ge 1$ such that
for any $n\ge n_\zs{0}$ there exists a  $\sigma$ - field $\cG_\zs{n}$
for which
 the random vector
 $\wt{\xi}_\zs{d,n}=(\xi_\zs{j,n})_\zs{1\le j\le d}$
 is the $\cG_\zs{n}$ conditionally Gaussian in $\bbr^{d}$ with the covariance matrix
\begin{equation*}\label{sec:Imp.6-1}
\G_\zs{n}=\left(
\E\,\xi_\zs{i,n}\,\xi_\zs{j,n}|\cG_\zs{n})
\right)_\zs{1\le i,j\le d}
\end{equation*}
and for some nonrandom
constant $l^{*}_\zs{n}>0$
\begin{equation*}\label{sec:Imp.6-1-0}
\inf_\zs{Q\in\cQ_\zs{n}}\quad
\left(
\tr \,\G_\zs{n}
-
\,
\lambda_\zs{max}(\G_\zs{n})
\right)
\geq l^{*}_\zs{n}\,,
\end{equation*}
where $\lambda_\zs{max}(A)$ is the maximal eigenvalue of the matrix $A$.
}

\noindent

\begin{proposition}\label{sec:Imp.Prop_2_1}
Let in the model \eqref{sec:In.1} the noise process describes by the L\'evy process \eqref{sec:In.1+1}. Then  the
condition $\D_\zs{1})$ holds with $l^{*}_\zs{n}=(d-1)\underline{\varrho}$  for any $n\ge 1$.
\end{proposition}

\begin{proposition}\label{sec:Imp.Prop_2_2}
Let in the model \eqref{sec:In.1} the noise process describes by the Ornstein--Uhlenbeck process \eqref{sec:Ex.1}. Then  the
condition $\D_\zs{1})$ holds with $l^{*}_\zs{n}=(d-6)\underline{\varrho}/2$  for any $n\ge n_0$ and
$d\ge d_0=\inf\{d\ge 7\,:\,5+\ln d\le \check{a} d\}$,
$\check{a}=(1-e^{-a_\zs{max}})/(4a_\zs{max})$.
\end{proposition}

Now,
for the first $d$ Fourier coefficients
 we use
 the improved estimation method proposed
for parametric models
 in
\cite{Pchelintsev2013}. To this end we
set $\wt{\theta}_\zs{n}=(\wh{\theta}_\zs{j,n})_\zs{1\le j\le d}$.
In the sequel we will use the norm $\vert x\vert^{2}_\zs{d}=\sum^{d}_\zs{j=1}\,x^{2}_\zs{j}$
for any vector $x=(x_\zs{j})_\zs{1\le j\le d}$ from $\bbr^{d}$.
Now
we define the shrinkage estimators as
\begin{equation*}\label{sec:Imp.12}
\theta^{*}_\zs{j,n}=
\left(1-g(j)\right)\wh{\theta}_\zs{j,n}\,,
\end{equation*}
where  $g(j)=(\c_\zs{n}/|\wt{\theta}_\zs{n}|_\zs{d}) \Chi_\zs{\{1\le j\le d\}}$ and
\begin{equation}
\label{coefficient_c_n}
\c_\zs{n}=
\frac{l^{*}_\zs{n}}{\left(r^{*}_\zs{n}+\sqrt{d/v_\zs{n}}\right)\,n}\,.
\end{equation}
The positive parameter
 $r^{*}_\zs{n}$  is such that $\lim_\zs{n\to\infty}\,r^{*}_\zs{n}\,=\infty$ and
for any $\epsilon>0$
\begin{equation}\label{sec:Imp.12+1_r_n}
\lim_\zs{n\to\infty}\,
\frac{r^{*}_\zs{n}}{n^{\epsilon}}
\,=\,0
\end{equation}
and $v_\zs{n}$ defined in \eqref{upsilon-nnn}. Now we introduce a class of shrinkage
weighted least squares estimates for $S$ as
\begin{equation}\label{sec:Imp.11}
S^{*}_\zs{\lambda}=\sum^{n}_\zs{j=1}\lambda(j)\theta^{*}_\zs{j,n}\phi_\zs{j}\,.
\end{equation}

We denote the difference of quadratic risks of the estimates \eqref{sec:Mo.1+??+} and \eqref{sec:Imp.11} as
$$
\Delta_{Q}(S):=\cR_\zs{Q}(S^{*}_\zs{\lambda},S)-\cR_\zs{Q}(\wh{S}_\zs{\lambda},S)\,.
$$
We obtain the following result.

\begin{theorem}\label{Th.sec:Imp.1}
Let the observed process $(y_\zs{t})_\zs{0\le t \le n}$ describes by the equation \eqref{sec:In.1} and the condition
$\D_\zs{1})$ holds. Then for any $n\ge 1$
\begin{equation}\label{sec:Imp.11+1}
\sup_{Q\in\cQ_\zs{n}}\,\sup_\zs{\Vert S\Vert\le r^{*}_\zs{n}}
\Delta_{Q}(S)\le-\c^2_\zs{n}
\,.
\end{equation}
\end{theorem}

\begin{remark}\label{Re;sec:Imp.1}
The inequality \eqref{sec:Imp.11+1} means that non asymptotically, i.e. for any $n\ge 1$,
 the estimate \eqref{sec:Imp.11}   outperforms in mean square accuracy the estimate \eqref{sec:Mo.1+??+}.
Moreover in the efficient weight coefficients  $d \approx n^{\check{\delta}}$ as $n\to \infty$ for some $\check{\delta}>0$.
Therefore, in view of the  definition  \eqref{coefficient_c_n} and the conditions
\eqref{sec:Ex.01-2}
 and
 \eqref{sec:Imp.12+1_r_n}
 $n \c_\zs{n}\to\infty$ as $n\to\infty$.
This means that improvement is considerably may better than for the parametric regression
when the parameter dimension $d$ is fixed  \cite{Pchelintsev2013}.
\end{remark}

\bigskip

\section{Improved model selection}\label{sec:IMo}

This Section gives the construction of a model selection procedure  for
estimating a function $S$ in \eqref{sec:In.1} on the basis of improved weighted least square estimates $(S^{*}_\zs{\lambda})_\zs{\lambda\in\Lambda}$ and states
the sharp oracle inequality for the robust risk of proposed procedure.

As in Section 3, the performance of any estimate $S^{*}_\zs{\lambda}$ will be measured by the
empirical squared error
$$
\Er_\zs{n}(\lambda)=\|S^*_\zs{\lambda}-S\|^2.
$$
In order to obtain a good estimate, we have to write a rule to choose a weight vector
$\lambda\in \Lambda$ in \eqref{sec:Imp.11}. It is obvious, that the best way is to minimise
the empirical squared error with respect to $\lambda$. Making use the estimate definition
\eqref{sec:Imp.11} and the Fourier transformation of $S$ implies
\begin{equation*}\label{sec:Mo.1+11+}
\Er_\zs{n}(\lambda)\,=\,
\sum^{n}_\zs{j=1}\,\lambda^2(j)(\theta^*_\zs{j,n})^2\,-
2\,\sum^{n}_\zs{j=1}\,\lambda(j)\theta^*_\zs{j,n}\,\theta_\zs{j}\,+\,
\sum^{n}_\zs{j=1}\theta^2_\zs{j}\,.
\end{equation*}
Here one needs to replace  the terms
$\theta^*_\zs{j,n}\,\theta_\zs{j}$ by their estimators
$\bar{\theta}_\zs{j,n}$. We set
\begin{equation*}\label{sec:Mo.2+=???}
\bar{\theta}_\zs{j,n}=
\theta^*_\zs{j,n}\,\wh{\theta}_\zs{j,n}-\frac{\wh{\sigma}_\zs{n}}{n}\,,
\end{equation*}
where $\wh{\sigma}_\zs{n}$ is defined in \eqref{sec:Mo.4+===}.
For this change in the empirical squared error, one has to pay
some penalty. Thus, one comes to the cost function of the form
\begin{equation*}\label{sec:Mo.4+++???}
J_\zs{n}^*(\lambda)\,=\,\sum^{n}_\zs{j=1}\,\lambda^2(j)(\theta^*_\zs{j,n})^2\,-
2\,\sum^{n}_\zs{j=1}\,\lambda(j)\,\bar{\theta}_\zs{j,n}\,
+\,\delta\,P_\zs{n}(\lambda)\,,
\end{equation*}
where $\delta$ is some positive constant and the penalty term
$P_\zs{n}(\lambda)$ is  defined  in \eqref{sec:Mo.6}.
Substituting the weight coefficients, minimizing the cost function
\begin{equation}\label{sec:Mo.6-1}
\lambda^*=\mbox{argmin}_\zs{\lambda\in\Lambda}\,J_n^*(\lambda)\,,
\end{equation}
in \eqref{sec:Imp.11} leads to the improved model selection procedure
\begin{equation}\label{sec:Mo.7}
S^*=S^*_\zs{\lambda^*}\,.
\end{equation}
It will be noted that $\lambda^*$ exists because
 $\Lambda$ is a finite set and also if the
minimizing sequence in \eqref{sec:Mo.6-1} $\lambda^*$ is not
unique, one can take any minimizer.

\begin{theorem}\label{sec:Mo.Th.1}
If the conditions $\C_\zs{1})$ and $\C_\zs{2})$ hold for the
distribution $Q$ of the process $\xi$ in \eqref{sec:In.1}, then, for any $n\geq1$ and $0<\delta<1/3$,
the risk \eqref{sec:risks_00} of estimate \eqref{sec:Mo.7} for $S$
satisfies the oracle inequality
\begin{equation*}\label{sec:Mo.10}
\cR_\zs{Q}(S^*_\zs{\lambda^*},S)\,\le\, \frac{1+3\delta}{1-3\delta}
\min_\zs{\lambda\in\Lambda} \cR_\zs{Q}(S^*_\zs{\lambda},S)
+\frac{\check{\B}_\zs{n}(Q)}{n \delta}\,,
\end{equation*}
where
$\check{\B}_\zs{n}(Q)=\check{\U}_\zs{n}(Q)
\left(
1+\vert\Lambda\vert_\zs{*}\E_\zs{Q}|\wh{\sigma}_\zs{n}-\sigma_\zs{Q}|
\right)
$
and
 the coefficient $\check{\U}_\zs{n}(Q)$ satisfies the property
\eqref{termB_rest}.
\end{theorem}

\vspace{2mm}

Now Theorem \ref{sec:Mo.Th.1} and Proposition \ref{sec:Mo.Prop.1} directly
imply the following inequality for the robust risk \eqref{sec:risks} of the procedure  \eqref{sec:Mo.7}.

\begin{theorem}\label{sec:Mo.Th.2}
Assume that the conditions $\C_\zs{1}^*)$ and
$\C_\zs{2}^*)$ hold and the function $S$ is
continuously differentiable.
 Then  for any $n\ge 2$  and
$0<\delta<1/3$
\begin{equation*}\label{sec:Mo.10+1-2}
\cR^{*}_\zs{n}
(S^*_\zs{\lambda^*},S)\,\le\, \frac{1+3\delta}{1-3\delta}
\min_\zs{\lambda\in\Lambda} \cR^{*}_\zs{n}\,(S^*_\zs{\lambda},S)
+\frac{\check{\U}^{*}_\zs{n}(1+\|\dot{S}\|^2)}{n \delta}\,,
\end{equation*}
where
 the coefficient $\check{\U}^{*}_\zs{n}$ satisfies the property
\eqref{termB_rest}.
\end{theorem}

\section{Asymptotic efficiency}\label{sec:Ae}

In order to study the asymptotic efficiency we define the following functional Sobolev ball
\begin{equation*}\label{sec:Ae.1}
W_\zs{k,\r}=\{f\in\C^{k}_\zs{p}[0,1]\,:\,
\sum_\zs{j=0}^k\,\|f^{(j)}\|^2\le \r\}\,,
 \end{equation*}
where $\r>0$ and $k\ge 1$ are
some unknown parameters, $\C^{k}_\zs{p}[0,1]$ is the space of
 $k$ times differentiable $1$ - periodic $\bbr\to\bbr$ functions
 such that $f^{(i)}(0)=f^{(i)}(1)$  for any $0\le i \le k-1$. It is well known that for any $S\in W_\zs{k,\r}$
 the optimal rate of convergence is
$n^{-2k/(2k+1)}$ (see, for example, \cite{Pinsker1981, Nussbaum1985}).
On the basis of the model selection procedure
we construct the adaptive
procedure $S_\zs{*}$ for which we obtain the following asymptotic upper bound
for the quadratic risk, i.e.
 we show that the parameter \eqref{sec:Ae.3} gives a lower bound
for the asymptotic normalized risks.
To this end we denote by $\Sigma_\zs{n}$ of all estimators $\wh{S}_\zs{n}$ of $S$ measurable with respect to
the process
\eqref{sec:In.1}, i.e.
$\sigma\{y_\zs{t}\,,\,0\le t\le n\}$.

\begin{theorem}\label{Th.sec: Ae.2}
The robust risk \eqref{sec:risks}
admits the following asymptotic lower bound
 \begin{equation*}\label{sec:Ae.5++==+??}
\liminf_\zs{n\to\infty}\,
\inf_\zs{\wh{S}_\zs{n}\in\Sigma_\zs{n}}
\,v_\zs{n}^{2k/(2k+1)}
\sup_\zs{S\in W_\zs{k,r}}\,\cR^{*}_\zs{n}(\wh{S}_\zs{n},S)
\,
\ge l_\zs{*}(\r)  \,.
 \end{equation*}
\end{theorem}

\noindent
We show that this lower bound is sharp in the following sense.

\begin{theorem}\label{Th.sec: Ae.1}
The quadratic risk \eqref{sec:risks} for the
 estimating procedure $S^{*}$ has the following asymptotic upper bound
 \begin{equation*}\label{sec:Ae.4}
\limsup_\zs{n\to\infty}\,v_\zs{n}^{2k/(2k+1)}
\sup_\zs{S\in W_\zs{k,\r}}\,\cR^{*}_\zs{n}(S^{*},S)
\,
\le l_\zs{*}(\r)
\,.
 \end{equation*}
\end{theorem}

\noindent
It is clear that Theorem \ref{Th.sec: Ae.1}
 and Theorem \ref{Th.sec: Ae.2}
imply
\begin{corollary}\label{Co.sec: Ae.1}
The model selection procedure $S^{*}$
is efficient, i.e.
\begin{equation}\label{sec:Ae.5+6+?}
\lim_{n\to\infty}\,(v_\zs{n})^{\frac{2k}{2k+1}}\,
\sup_\zs{S\in W_\zs{k,\r}}\,\cR^{*}_\zs{n}(S^{*},S)\,
= l_\zs{*}(\r)
\,.
\end{equation}
\end{corollary}

\noindent
Note that the equality
\eqref{sec:Ae.5+6+?} implies that the parameter
\eqref{sec:Ae.3} is the Pinsker constant
in this case (cf. \cite{Pinsker1981}).
Moreover, it  means that the robust efficiency holds with
the  convergence rate $(v_\zs{n})^{\frac{2k}{2k+1}}$.
It is well known that for the simple risks
 the optimal (minimax) estimation convergence rate
for the functions from the set  $W_\zs{k,r}$
 is $n^{2k/(2k+1)}$ (see, for example, \cite{IbragimovKhasminskii1981, Nussbaum1985, Pinsker1981}).
So,
 if  the distribution upper bound  $\varsigma^{*}\to 0$  as $n\to\infty$
 we obtain the more rapid rate, and
 if $\varsigma^{*}\to \infty$  as $n\to\infty$
we obtain the more slow rate. In the case when $\varsigma^{*}$ is constant the robust rate is the same as the classical non robust convergence rate.

\bigskip

\noindent
{\bf Acknowledgements.}

The first author is partially supported by the
 the RSF grant 17-11-01049.
The last author is partially supported  by the Russian Federal Professor program (project no. 1.472.2016/1.4,
  Ministry of Education and Science) and
  by the project XterM - Feder, University of Rouen, France.

\bigskip

redpajama_set_name: RedPajamaArXiv


William R. Allen Road, also known as Allen Road and The Allen is a short expressway and arterial road in Toronto, Ontario, Canada. It starts as a controlled-access expressway at Eglinton Avenue West north to just south of Transit Road, then continues as an arterial road north to Kennard Avenue, where it continues north as the northern portion of Dufferin Street. Allen Road is named after late Metro Toronto Chairman William R. Allen and is maintained by the City of Toronto. Landmarks along the road include the Lawrence Heights housing project, Yorkdale Shopping Centre and Downsview Park (formerly CFB Downsview). A section of the Toronto subway Line 1 Yonge–University is located within its median from Eglinton Avenue to north of Wilson Avenue, and briefly runs parallel to it underground to the vicinity of Sheppard Avenue.
The portion south of Sheppard Avenue was originally constructed as part of the Spadina Expressway project. The Spadina Expressway was a proposed north-south freeway, intended to connect downtown Toronto to the suburb of North York, and to serve the Yorkdale Shopping Centre project. It was only partially built before being cancelled in 1971 due to public opposition. It was proposed in the mid-1950s as part of a network of freeways for Metro Toronto. Its cancellation prompted the cancellation of the rest of the network.
Allen Road, looking north from Glencairn Avenue.
Allen Road at Eglinton Avenue West. A 1 m (3 ft 3 in) strip of land was purchased south of Eglinton by the City of Toronto, preventing the expressway from being extended.
The road begins at Eglinton Avenue West with two separate signalized intersections with the street. The northbound lanes intersect with Eglinton to the east of the Eglinton West subway station, and the two southbound lanes connect to Eglinton Avenue west of the station. The road proceeds north to Lawrence Avenue West as a four-lane freeway with a speed limit of 80 km/h (50 mph) in the northerly direction and 80 km/h until 500 m (1,600 ft) north of Eglinton Avenue. The tracks of Line 1 Yonge–University are situated between the roadways. There is a subway station at Glencairn Avenue. The roadway and subway are situated in a cut-out section of land, with grass and trees on either side.
The roadway intersects with Lawrence Avenue West with on- and off-ramps which are signalized. The Lawrence West subway station and bus terminals are located between the two roadways. From Lawrence Avenue north, the roadway is six-lanes, with a speed limit of 80 km/h. The Allen connects to Yorkdale Road by on- and off-ramps. The Yorkdale subway station is located between the two roadways, and over the Yorkdale connecting ramps. For a short stretch, the Allen is eight lanes, the lanes connected to the Yorkdale Road exits. On either side of the road through this stretch concrete retaining walls were built on both sides.
The interchange of the Allen and Highway 401 is a hybrid of the turbine and clover-stack. The Allen proceeds on bridges over the lanes of the 401, with ramps from the Allen to 401 passing overhead. The interchange also serves to connect Yorkdale Road with the 401. The exit from the northbound Allen to the 401 serves as a ramp to both directions of the 401, with two lanes proceeding north of the 401. Similarly, two lanes are provided southbound over the 401, and access from the eastbound and westbound 401 merges with the Allen south of Yorkdale Road. Access to Yorkdale Road from the east-bound 401 is provided at the partial Dufferin Street interchange to the west. The southbound ramp from the Allen to the eastbound 401 flies over the whole interchange and connects with the eastbound 401 collector lanes. The interchange is one of the few on Highway 401 that uses conventional light poles instead of high-mast lighting due to nearby Downsview Airport. The subway is situated on bridges over the 401 between the northbound and southbound lanes.
From north of the 401, the Allen is four or six lanes, passes over Wilson Avenue (with the Wilson subway station being above the namesake avenue within Allen Road's median), and meets with Transit Road at a signalized intersection. The subway diverges from the route just north of Wilson Avenue, with a large subway yard and bus garage, known as Wilson Yard, to the west of the road. Just south of Transit Road, a partial interchange with Wilson Heights is provided. It was the former northern terminus of the road until 1982. North of Transit Road, Allen Road is an arterial road of four- or six-lanes, meeting Sheppard Avenue West at a signalized intersection. The subway line, now underground, returns to follow the road's alignment just south of Sheppard West subway station on the east side of the intersection, with the bus terminal being on the southeast corner of Sheppard. The line continues north of the road under a grass berm, then curves west to cross under it and leave the corridor. Allen Road's speed limit through that section is 70 km/h (43 mph). It continues north, still maintaining the higher speed limit with signalized intersections at Rimrock and Kennard Avenue. The road becomes Dufferin Street north of Kennard Avenue.
The roadway is lit by high pressure sodium with shaded luminaires on cobra-neck poles. The original installation was low-pressure sodium lighting in 1969. It was the first installation of low pressure sodium lighting on existing cobra-neck poles in Toronto. This was later introduced to the Don Valley Parkway and Gardiner Expressway. In 1991, the Allen became the first municipal expressway in Toronto to have its lighting converted to high pressure sodium with shaded luminaires. This was done using the existing cobra-neck style of poles unlike the Don Valley and Gardiner where a combination of high mast and new conventional poles was introduced.
The Spadina Expressway was intended to run from north of Highway 401 into the downtown of Toronto via the Cedarvale and Nordheimer Ravines and Spadina Road. Various versions of the proposal showed it starting to the north of North York at today's highway 407, between Bathurst and Dufferin Streets. It then travelled south to meet highway 401 a half-mile east of Dufferin.
The interchange at the 401 evolved over time to a complex 26 bridge interchange, the most-complex in Ontario to that time. The Spadina was envisioned at first to provide access to the Yorkdale development from the 401, with a four-lane road. While the Spadina was being planned, the 401 highway was being developed into the present-day 12-lane highway and the Spadina was being revised to a six-lane highway to downtown, with a subway line in the middle. It cost $13 million in 1960s dollars to build, after being initially estimated to cost $1 million.
From the 401 south to Eglinton, the roadway was to be in a trench, with the rapid-transit line in the middle. South of Eglinton, it continued into the Cedarvale Park below ground level. The route south of Eglinton was never built and several variations of the plan were proposed. Plans were initially for the road to be on the surface of the ravines and the subway below the surface, at least as far as Spadina Road with a tunnel under St. Michael's College north of St. Clair. Another plan projected the roadway to be completely underground also for this stretch, on top of the subway line. From Spadina Road south of St. Clair and further south, the roadway would be underground to Davenport Road, opening into an interchange on Davenport Road.
The stretch from Davenport south was also not built. 'No highway' routes proposed rebuilding Spadina Road and Madison Avenue south to Bloor Street as arterial roads. The highway plan proposed a highway width from Spadina's west side to Madison's east side, in an exposed trench south to Bloor Street. If a Crosstown Expressway were to be built then the Davenport interchange might not have been built and a Crosstown interchange instead.
Various proposals for the Spadina south of Bloor Street were made. The 1969 functional design proposed an express route in the centre of the Spadina Avenue, and parallel two-lane streets on either sides to provide access to the businesses. Other proposals included no highway south of Sussex Street, just north of Harbord.
By the 1940s, urban development extended past the City of Toronto's borders. It was recognized within the planning department of the city that population growth would take place and that the farmlands outside of the City's border would be developed. In 1943, the City of Toronto Planning Board developed a plan for the area within a nine-mile radius of Yonge Street and Queen Street. It included a network of superhighways that included the Spadina Expressway.
In November 1947, the Toronto City Planning Board presented a plan to the City of Toronto Civic Work Committee for two new arterial roads: one along the lake shore running east-west and another running north-south to the west of downtown. The lake shore route was at first abandoned, while the north-south route was approved. The Spadina Road project would be a new road from Front Street at the south, to St. Clair Avenue to the north, along the route of the existing Spadina Avenue and Spadina Road. The "jog" at Bloor Street connecting the two existing roads would be straightened, the existing Spadina Road widened, and a new cut of the Davenport Road escarpment would be made, taking the road to St. Clair Avenue. The proposal was added to the January 1, 1948 municipal election, where it was approved narrowly by the voters by a vote of 34,261 to 32,078. While the proposal was adopted, the narrow approval led councillors of the time to hold off on approval of construction.
A proposal for a highway from the north-west to downtown was developed in 1949 by the Toronto and Suburban Planning Board, part of a plan for numerous expressways in the Toronto area, including the "Lakeshore Expressway" (the eventual Gardiner Expressway) and Don Valley Parkway highways. It was initially to be named "North West Drive", or "Spadina Road Extension". The route was laid out by two members of the board, future Metro chairman Fred Gardiner and James P. Maher, chairman of the Toronto Planning Board. The proposal died when York Township rejected the idea.
When Metropolitan Toronto (Metro) was formed in 1954, one of its first priorities was highway building. Metro proposed building routes into and out of downtown, as well as encircle the downtown with an "expressway ring." The routes of the Lakeshore and Don Valley expressways were less controversial and allowed to proceed, while others such as the Crosstown, Scarborough and Spadina Expressways were put off for further study. The other expressways planned to cut through developed areas, and were also considered to be needed less urgently. A parallel development to extend the provincial Highway 400 was proposed to the west, to connect to the Lakeshore Expressway in the area of Fort York.
In 1959, the Spadina Expressway became part of the Metro official transportation plan. The original plan intended to connect a "Highway 403 bypass" in the vicinity of today's Highway 407 in the city of Vaughan south through the borough of North York, just east of Downsview airport, then south between Dufferin Street and Bathurst Street as far south of Eglinton. The highway would have gone into a ravine as far south as St. Clair Avenue through the borough of York. It would then enter Toronto proper, going directly south through the Annex neighbourhood, connecting to an east-west "Crosstown Expressway" south of Dupont Street, and ended at the intersection of Bloor Street and Spadina Avenue. Spadina Avenue would be reconstructed with express lanes in the middle all the way south to the waterfront.
The project planned a rapid-transit line in conjunction with the expressway. It would operate above ground north of Eglinton Avenue, and travel underground south to connect at Spadina station of the Bloor-Danforth line. The above-ground section was situated between the lanes of the highway.
The expressway was considered critical for the development of the planned $42 million CAD Yorkdale Shopping Centre, south-west of the Spadina-401 interchange. At one point, the Yorkdale development was under the threat of cancellation without the approval to proceed with the Spadina. Only after Metro Council formally approved the whole Spadina project in 1962, did the land owners T. Eaton Co. Limited and developers Webb and Knapp (Canada) Limited announce construction. The interchange of the 401 with today's William R. Allen Road has connections with the private roadways of Yorkdale.
Estimates of the cost were first determined in 1961, of $65 million CAD, with construction to proceed from 1967 until 1970. At that time, Metro and Toronto were in discussions about the route south of Dupont Street, as to whether it would be an elevated highway or at ground level in that section.
On December 12, 1961, Metro Council first approved the Spadina Expressway project, committing $5 million CAD in a 13–8 vote. This covered the cost of the first section from Highway 401 to Lawrence Avenue. At the same time Council put off approving the whole route and voted 19–2 to delete the Crosstown Expressway from the transportation plan.
The plan proposed the most-complex highway interchange attempted in Ontario to that point. It covers 40 hectares (99 acres).
Opposition to the project started before construction began. In 1960, members of the Cedarvale Ratepayers Association disrupted meetings of the Metro Toronto Roads Committee discussing the project. York Township, which became the Borough of York, opposed the construction of the highway through its municipality, and through the York Township-owned Cedarvale Ravine, characterized as "the only park area west of Bathurst Street and north of St. Clair Avenue available to serve 100,000 citizens", and members of the Association proposed a study of the need for the expressway, and to suggest studying the route of Dufferin Street instead. The Roads Committee turned down their requests. York Township threatened to go to the Supreme Court of Canada to block Metro from taking their park. Metro Chairman Fred Gardiner opined, "I can't see how anyone would allow one of 13 municipalities to block an expressway."
As the route of the proposed Crosstown Expressway was being drawn up at the same time, neighbouring residents of Rosedale opposed the construction of the Crosstown Expressway intended to connect to the Spadina, through their neighbourhood. Routing of the Crosstown was proposed along Dupont Street in the area, although the City of Toronto proposed a routing north of the railway lines, closer to Davenport. The Crosstown would continue east through the Rosedale neighbourhood to connect to the Don Valley Parkway.
In 1961, Metro Roads Committee held meetings to hear submissions on the routing of the expressway. Forest Hill Village objected to the proposed route of the expressway though the village, as the village would suffer "serious economic loss" according to Reeve Laurie Simonsky. The road and the interchange at Eglinton Avenue would require the demolition of 276 buildings and bisect the village. Forest Hill proposed a tunnel from the Cedarvale Ravine north, under Forest Hill. Gardiner, former reeve of Forest Hill, admitted that the project would be harmful to the village, "but there is urgent need for an expressway to serve the northwest Metro area" and that the route through the village was the only one that would allow the expressway to enter the Cedarvale Ravine.
In June 1961, the section of the Spadina Expressway south of the Crosstown Expressway was cancelled. The Spadina would now terminate at an interchange with the Crosstown, and Spadina Road north of Bloor Street would be widened. This ended a dispute between the City of Toronto and Metro Toronto. Metro wanted to build an expressway through to the Gardiner, while the city wanted to build an expressway further west, in the vicinity of Christie Street, which would connect the Crosstown to the Gardiner. The plan to build down Spadina Avenue would have demolished 1 Spadina Crescent, in the centre of Spadina, just north of College Street.
Opponents to the expressway started organizing. Ratepayers Associations banded together to object to the municipal expressway plan, forming the Coordinating Committee of Toronto Ratepayers Associations and the Metro Ratepayers Transportation Committee. The ratepayers objected to the $400 million CAD municipal expressway plan as expensive and "unleashing a torrent of private passenger vehicles into the city centre." University of Toronto professor James Acland of the Rathnelly Residents Association spoke of the futility of combining rapid transit and expressways on one route. "They won't persuade anyone to park his car and take rapid transit when there is a wonderful expressway inviting him to drive downtown." S. A. Hudson, president of the Lawrence Heights Ratepayer Association cited figures showing the roadway would carry 10,000 vehicles into the core at rush hour, requiring 69 acres (28 ha) for parking alone. The group placed ads in newspapers prior to the December 12, 1961 vote of Metro Council on the Spadina and Crosstown expressways, urging the rejection of the plan. The pressure was partially effective as Council voted 19–2 to delete the Crosstown, but approved the first stage of the Spadina by 13–8, while putting off approval of the downtown route.
The vote put the whole project in doubt. At the time the province paid for half of the cost of roadways, but it did not contribute to rapid transit. The province before approving any road funds for the project, wanted Council approval for the whole project, while the vote covered only the Lawrence to Highway 400 section, including the interchange. The province's transportation minister, William Goodfellow, wrote to Metro Council to state that since Metro had not voted to approve the whole project, that the province would not consider connecting Spadina with Highway 401.
The vote to have Metro's Road Committee study the route south of Lawrence, led to the Roads Committee to hold public hearings. North York Council voted unanimously to fight for approval of the whole project. Councillor Irving Paisley blamed that downtown business interests were behind opposition to the project. "The whole scheme is being jeopardized by several organizations with political strings, local grievances or selfish aims." Paisley organized a campaign to support the building of the project, together with the Yorkdale developer Webb and Knapp. Paisley himself wrote the submissions of 8 of the 25 North York ratepayer associations in favour of the project, including the submissions of some associations that were defunct. Employees of the Yorkdale developer started a letter-writing campaign, and Paisley and Webb and Knapp collaborated on submissions.
Hearings heard deputations from more than 30 ratepayer associations. Strong support was heard from North York associations and opposition was mostly from York, Forest Hill and the City of Toronto associations. Opponents also proposed a $10 auto tax and $25 truck tax to pay for the cost of Metro expressways instead of paying the costs from property taxes and wanted Metro to finish the Gardiner and Don Valley expressways before starting any others. York Reeve William Saunders became a staunch opponent of the project, publicly announcing that York would fight the project in court. York's Cedarvale Park was in the path of the project and by law, the lands could not be taken by Metro without York's consent. The Roads Committee asked the Metro Legal Committee to look into proposing provincial legislation to get the lands.
On February 19, 1962, Metro Roads Committed approved the whole project by a 5–1 vote, the only dissenter being future Toronto Mayor William Dennison. The meeting was picketed by opponents with signs proclaiming "Spadina Expressway No!", "Taxes at Critical Level" and "We are Watching How You Vote." The committee also recommended removing the Crosstown from the plan. Metro Chairman William R. Allen, whom the road would ultimately be named after, spoke in favour of the project based on the rapid transit portion of the project, which included commuter parking lots at northern stations. "If this does not get the motorist out his vehicle and back to rapid transit, Metro Council cannot be blamed."
the addition of more cars to the downtown.
By 1963, costs had risen to over $73 million CAD for the plan. Metro, which was also constructing the Gardiner Expressway, Don Valley Parkway and Bloor-Danforth subway lines had fallen under the scrutiny of the Ontario Municipal Board (OMB) over its spending. The OMB had to approve the 1963 budget before Metro could. The Spadina was separated from the 1963 budget, and the OMB held hearings into the project. In submissions to the OMB, the townships of Forest Hill and York again objected. CCTTRA, and the CCTRA noted its objections. Allan Ackman, of the Wellesley-Bloor Street ratepayers asked "what compensation is there for all the people will be exposed to the deadly poisonous clouds of fumes from the cars on it?" The Cedarvale ratepayers had obtained the advice of Lewis Mumford on matters of town planning and stated that the expressway was of no use without the cancelled Crosstown. Publisher Allan Perly objected to Yorkdale being such a beneficiary. The Deer Park ratepayers objected on the increases of taxation. The OMB upheld the Spadina project. The OMB stated in its decision that the "sectional interest must give way to the public need of the larger area." On the issue of the ravine parklands, the OMB stated "The board should and does expect that any park land that may be lost to York Township as a result of this undertaking will be replaced, insofar as may be possible in the circumstances, by suitable alternative lands for that purpose."
Construction started in 1963 with the clearing of the route. The area north of Lawrence Avenue was open land. South of Lawrence, dozens of homes were demolished. Coinciding with the opening of the Yorkdale mall, an interim roadway was opened from Lawrence north to Yorkdale Road in 1964.
In 1964, Metro released another transportation plan, which extended the Spadina south of Bloor, again requiring the demolition of homes south of Davenport. Toronto City Council adopted an Official Plan opposing the Crosstown Expressway and the Christie Expressway completely. Ontario's Minister of Municipal Affairs overruled the city, and modified the city's Plan to allow for the construction of both expressways. The City and Metro were now in disagreement.
In 1966, the section from Lawrence Avenue north to Wilson Avenue opened. Construction then started on the section south to Eglinton Avenue. York Council had dropped its opposition to the expressway and made an agreement with Metro on the use of Cedarvale Park for the expressway. This agreement provided for the creation of 12 acres (4.9 ha) of park lands in the Borough of York to replace the park lands lost to the expressway trench. This plan would have meant the expropriation of homes for the replacement lands and residents of York protested the plan to the Council. The cost of the expropriation plan was an estimated $4 million of construction, plus the loss of the assessment, while putting a cover over the roadway within the park would have cost $5 million. The opposition led Metro to agree to building the expressway within a tunnel under the park.
As construction proceeded, opposition to the expressway grew among City of Toronto residents. In October 1969, the "Stop Spadina, Save Our City Co-ordinating Committee" (SSSOCCC) was formed, under the chairmanship of University of Toronto professor Alan Powell. The group was a coalition of students, academics, politicians, ratepayer groups and business people. Notable among the opposition was urban theorist Jane Jacobs, who moved to the Annex in 1969, fresh from a battle to stop the Lower Manhattan Expressway in New York City. Marshall McLuhan, too, was opposed to the expressway and said: "Toronto will commit suicide if it plunges the Spadina Expressway into its heart... our planners are 19th century men with a naive faith in an obsolete technology. In an age of software Metro planners treat people like hardware — they haven't the faintest interest in the values of neighbourhoods or community. Their failure to learn from the mistakes of American cities will be ours too." In the 1969 civic election, three councilors were elected in Toronto on a platform of immediately ending Spadina construction: Ying Hope, William Kilborn and John Sewell.
By 1969, all but $10 million of the approved $76 million was spent, completing the roadway only to Lawrence Avenue, and the road bed to Eglinton Avenue. Metro learned that the project would require a further $80 million for completion and halted construction and decided to review the project. The total cost of the project (including the rapid transit line) was now $237 million. A trench had been dug in Cedarvale Park, and Metro Roads and Traffic Commissioner Sam Cass attempted to commit Metro to construction south of Eglinton by arranging to call for tenders in building a tunnel in the park. The call, going against Metro Council's explicit instructions, was noticed only one day before they would be published. The call was cancelled by Metro Chairman Albert Campbell.
SSSOCCC developed its public campaign. SSSOCCC produced a short film by McLuhan entitled "The Burning Would" explaining the reasons to stop the project while poking fun at expressway backers. SSSOCCC also held public lectures with Jacobs and started a petition campaign. SSSOCCC members David and Nadine Nowlan, professors at U of T released their book The Bad Trip, an economic analysis of the project and explanation of their opposition.
Metro Council voted to apply to the OMB for permission to borrow the funds and requested that the OMB held hearings. OMB Hearings began on January 4, 1971. Opposition groups banded together under the banner of "The Spadina Review Corporation" and hired one of Canada's top trial lawyers, John Josiah "J. J." Robinette, to plead their case. Metro presented its case based on technical studies showing the road was needed to manage expected traffic. Council was represented by its solicitor and its witnesses included Metro and City Commissioners and American transportation planner Alan Voorhees. Opposition groups based their case on the factors of noise, pollution, destruction of homes and the expected increase of traffic the roadway would cause. Their witnesses included Jack Fensterstock of the New York City Department of Air Resources, neighbourhood residents, as well as urban planners, economists and architects. No elected officials, nor the Metro chairman, appeared to defend or oppose the project. The Board held 16 days of hearings and gave its approval by a vote of 2–1, OMB chairman J. A. Kennedy dissenting, on February 17, 1971.
Toronto Mayor Dennison was shocked at the result. "It's shocking that a group who never at any time suggested workable alternative routes has successfully opposed something as important in the growth of Metro as was the Don Valley". Metro Toronto chairman Albert Campbell was incensed at the provincial government, stating, "It may mean that we will never build another expressway."
The debate over the Spadina Expressway, and its eventual cancellation, is regarded as a watershed moment in local politics. Toronto City Council was changing at the time to oppose the "top-down" planning of the Metro government. A "Reform Era" in Toronto politics was beginning, which brought to Toronto City Hall David Crombie, John Sewell, Allan Sparrow and Colin Vaughan. This new council viewed the Metro government and its officials with suspicion as not being accountable to local residents. In the 1950s and 1960s, Metro and City Councils had pushed through numerous large projects in transportation, and housing. The impacts on the central neighbourhoods had been substantial and had led to grassroots organizing. Councillors Sewell and Vaughan came directly from the grassroots campaigns.
Who plans? – Until this point, planning had been done by professional planners or Metro department heads based on technical issues, such as projected traffic congestion. Metro had not provided a policy for the planners to follow.
Role of the OMB in policy – The OMB was in charge of approving capital borrowing, a consideration dating from the days of the Depression. In the case of the Spadina, it had had to decide on an issue far beyond approving whether a municipality could afford the project.
Role of the OMB in planning – The OMB became an approver of land development disputes, which often pitted municipalities or residents or developers against each other. Was this an appropriate role for the OMB?
Role of the Metro Chairman – The Metro Chairman did not appear at the OMB to defend the project at the OMB. Chairman Campbell took a neutral position on a very important project.
Premier Davis called a provincial election not long after the decision, in October 1971. Davis' campaign strategy used the Spadina decision to differentiate his government from past Progressive Conservative governments. Davis, who was both attacked and lauded for the Spadina cancellation, was re-elected with increased support in Toronto. Davis would remain in power in Ontario until 1985, when he retired from politics.
The cancellation marked the beginning of the end of construction of Metro's planned expressway network. Metro highway projects such as the Crosstown, Scarborough and Richview expressways did not proceed. Other than extending the Allen Road, Metro Toronto did not build another expressway, and erased the other expressways from its official plans. Metro would proceed in future years to complete the Spadina subway line, extend the Yonge Street line, and build the Sheppard subway line.
In the opinion of critics, the move set "in motion a trend that has yet to abate: politicians overruling the painstaking work of urban and transportation planners". There is often a long line of vehicles in the southbound lanes from as far north as Lawrence Avenue queued to exit on Eglinton and often long lineups of cars on Eglinton trying to get on the Allen Road. In addition, the Spadina subway line which is located within the median, has been criticized as the "worst place to put that route...The stations that serve the Allen Road corridor are among the most uninviting, unwalkable places in the city".
At the time of cancellation in 1971, the expressway was paved to Lawrence Avenue while the portion running further south to Eglinton Avenue had been graded only and was given the nickname the "Davis ditch". Traffic from and to the southerly end of the road at Lawrence, spilled onto neighbourhood streets, as the activists predicted, especially Marlee Avenue. Esther Shiner, who lived near the Lawrence intersection, was elected to North York Council in 1973 on a platform to get the expressway completed to Eglinton Avenue. She headed the "Go Spadina" public campaign that was successful in persuading Metro, against the wishes of the City of Toronto, to pave the ditch and opened the road to Eglinton on September 8, 1976.
In response, Davis made plans to transfer a strip of land south of Eglinton to the City of Toronto to block any further extension. Metro and the province ended their dispute in an agreement to build the Black Creek Drive arterial road, a southerly extension of Highway 400. Metro would transfer its Spadina lands south of Eglinton to the province, and the province would build Black Creek Drive south to Weston Road. Metro officials dragged their feet by attempting to get the buffer strip moved to Bathurst and St. Clair, enabling a possible future extension to Bathurst, and a widened Bathurst street, but the province threatened to simply expropriate the lands and the lands were turned over to the province in 1984.
On February 7, 1985, on his final day in office, Davis delivered to the City of Toronto a 1 metre (3.3 ft) wide strip of the land on the south side of Eglinton Ave. West at the Allen intersection, with a 99-year lease, blocking any possible extension to the south. Opponents such as Shiner had wanted the province to hold onto the land, hoping that a future premier would be willing to consider the highway. Shiner felt that "the expressway will be built, bit by bit, into the city". Shiner had received a $20 million estimate from Metro officials to extend Spadina as a four-lane south to Davenport.
After the land transfer, North York Council made several attempts to get Premier David Peterson to reconsider Davis's actions but he refused to meet Council representatives over the issue. Shiner attempted to get a Metro-wide plebiscite but failed. North York sponsored a telephone survey of Toronto residents, although a majority supported transit improvements instead. Shiner's attempts ended only when she died of cancer in 1987. Metro Chairman Dennis Flynn and Metro planners still pushed for the completion as late as 1988, with the release of a traffic study of northwestern Metro that recommended extending the Spadina south, but Metro Council defeated further studies in a 14–12 vote on July 5, 1988. Another proponent left the scene that year as long-time Metro Commissioner of Roads and Traffic Sam Cass retired.
The Spadina subway line was built in the median of the project right-of-way from Wilson to Eglinton. The route south of Eglinton follows the approximate route planned for later sections of the expressway, albeit underground. During the planning of the route, alternate routes directly south along Christie Street and south along Bathurst Street were also considered. After the use of the University section of the subway by Bloor-Danforth trains was abandoned, the original route of the subway to connect with the St. George station was agreed upon. The University section of the Yonge line was made into a part-time route until the Spadina subway extension was built. After the cancellation of the Expressway in 1971, the subway's construction had not been started. Route studies occurred again and the original route confirmed once more and it proceeded to construction. It opened in 1978.
In 1995, most of the 401-Allen interchange overpasses were rehabilitated. The interchange was later modified from 2001 to 2004, when Ontario widened the westbound collectors to four lanes, eliminating the forced exit lane.
In 1996, Metro Council voted to end the matter finally and sell the 112 expropriated properties south of Eglinton Avenue. The properties were appraised and sold at fair market value, offered first to their former owners. The proceeds were divided between Metro and Ontario, with Metro keeping two-thirds up to $30 million, and proceeds above $30 million split equally. One home purchased by Metro in 1967 for $50,000 had appreciated in value to $440,000 by 1997.
During the 2010 Toronto mayoral election, Rocco Rossi proposed completing the expressway in a tunnel to meet the Gardiner Expressway. However, Rossi's position in favour of extending the expressway conflicts with the plans for the Lawrence Heights revitalization project. Later that week, after much criticism of Rossi by other candidates and the media, Rossi revised his position to one of "studying" building a tunnel.
In 2012, the City of Toronto started an Environment Assessment of improvements to the Allen Road area. The study's goals include how to make Allen Road better for the people that travel on it and the communities that surround it. It says that "Continuing Allen Road south, similar to the original 1950s expressway plan, is not being considered an option."
In 2012, the City modified the Lawrence Avenue on-ramp to the north-bound Allen, removing the right-turn channelization and one-lane merge into a two-lane on-ramp for both east and westbound traffic. The southbound Allen off-ramp to Lawrence Avenue was converted to two lanes, and the lanes were changed to one lane for each of left and right turns.
Maintain the roadway as is.
Maintaining the roadway and making conditions for pedestrians easier and enhancing access to subway stations.
Making significant improvements to the corridor through the addition of HOV lanes, bike paths and pedestrian infrastructure.
Transform the roadway to a surface roadway.
Transform the roadway to a tunnel or deck.
Eliminating the roadway entirely and opening up the space for other uses. Subway service would remain in place.
A trailblazer to Allen Road.
The following table lists the interchanges along Allen Road. Like the Don Valley Parkway and Gardiner Expressway, exits along Allen Road are not numbered and are identified by street name. The entire route is located within Toronto (excluding driveway to Dufferin Street stub). Wilson Heights is the last northbound interchange with all other junctions being intersections with traffic signals.
0.0 Eglinton Avenue West Half diamond interchange – Northbound entrance and southbound exit.
North of Kennard Avenue, Allen Road becomes Dufferin Street. Dufferin Street runs semi parallel with Allen Road south from Kennard. This stub section of Dufferin is a single lane residential street from Sheppard Avenue West, ending at a cul-de-sac north of Clifton Avenue.
^ a b Osbaldeston 2008, p. 144.
^ "Two City Highways Will Cost $9 Million". The Globe and Mail. November 13, 1947. p. 1.
^ "Spadina Widening Must Go Through, Mayor Asserts". The Globe and Mail. January 6, 1948. p. 5.
^ "3 Controllers Favor Delaying Spadina Plan". The Globe and Mail. January 5, 1948. p. 1.
^ Colton 1980, p. 63.
^ Colton 1980, pp. 165–172.
^ "Start $42 Million Yorkdale Centre". The Globe and Mail. p. 27.
^ "Road Budget Emphasizes Expressways". The Globe and Mail. February 7, 1961. p. 1.
^ a b Baker, Alden (December 13, 1961). "Metro Discards Crosstown Plan". The Globe and Mail. p. 1.
^ a b "Citizens Opposing Expressway Give Gardiner Hard Time". The Globe and Mail. June 14, 1960. p. 5.
^ "Seek Authority to Investigate Spadina Route". The Globe and Mail. June 28, 1960. p. 5.
^ a b "Propose Tunnel Under Forest Hill For Part of Spadina Expressway". The Globe and Mail. April 18, 1961. p. 1.
^ "New City Predicted Near Toronto Township". The Globe and Mail. June 22, 1961. p. 5.
^ "Chaos Feared if Expressways Slice Up Metro". The Globe and Mail. December 6, 1961. p. 5.
^ "Road Chief Misinformed, Goodhead says". The Globe and Mail. January 13, 1962. p. 5.
^ "North York to Fight For Spadina Scheme". The Globe and Mail. January 16, 1962. p. 5.
^ Haggart, Ron (February 26, 1962). "The Brilliant Campaign to Make Public Opinion". Toronto Star.
^ "The Strange Case of the All-Alike Letters". Toronto Star. March 1, 1962.
^ a b "Want 14 Acres For Spadina Plan". The Globe and Mail. February 6, 1962. p. 5.
^ "Ask City Car, Truck Tax". The Globe and Mail. February 6, 1962. p. 5.
^ "Will Go to Courts To Block Expressway, York Reeve Threatens". The Globe and Mail. February 5, 1962. p. 5.
^ "Spadina Expressway, Rapid Transit Passed by Committee". The Globe and Mail. February 10, 1962. p. 5.
^ "Metro Approves Spadina Expressway, Transit Line". The Globe and Mail. March 7, 1962. p. 5.
^ "Spadina Expressway". The Globe and Mail. March 9, 1962. p. 6.
^ "Foes Attack Spadina Plan". The Globe and Mail. February 2, 1963. p. 27.
^ "OMB Approves Expressway Despite Protest". The Globe and Mail. August 9, 1963. p. 4.
^ Osbaldeston 2008, p. 146.
^ a b c Cherry, Zena (January 17, 1970). "Expressway group plans to SSSOCCC it to Toronto". The Globe and Mail. p. 11.
^ Osbaldeston 2008, p. 147.
^ a b Rose 1972, p. 139.
^ MacKenzie, James (January 26, 1970). "How the 20-year political nightmare of the Spadina Expressway happened". The Globe and Mail.
^ Dzeguze, Kaspar (October 16, 1970). "Burning Would comic rebuke of expressway backers". The Globe and Mail. p. 15.
^ Rose 1972, p. 138.
^ Rose 1972, pp. 138–139.
^ Rose 1972, p. 142.
^ The Globe and Mail, 1971-01-21, James MacKenzie, "Pollution Predicted to be worse than New York's", Toronto.
^ The Globe and Mail, 1971-01-22, James MacKenzie, "Monoxide level safe on present Spadina, expert tells OMB", Toronto.
^ Glynn, Dennis (June 4, 1971). "Dennison shocked by decision". The Globe and Mail. p. 1.
^ "'May never build another expressway': Campbell". The Globe and Mail. June 4, 1971. p. 1.
^ Rose 1972, pp. 140–143.
^ Rose 1972, p. 143.
^ a b c "North York politician earned wide respect". The Globe and Mail. December 21, 1987. p. A19.
^ a b Osbaldeston 2008, p. 149.
^ "York Protests But Spadina Ditch Opens". Metro News / Editorials. The Toronto Star. September 8, 1976. p. B1.
^ Stead, Sylvia; Baker, Alden (December 14, 1984). "Title to 3-foot strip Spadina assurance on way, Davis says". The Globe and Mail. p. 1.
^ Monsebraaten, Laurie (July 6, 1988). "Metro refuses to re-examine Spadina expressway proposal". Toronto Star. p. A7.
^ "Roads commissioner Sam Cass retiring after 34 years in Metro". Toronto Star. October 5, 1988. p. A6.
^ Black, Debra (September 18, 1997). "Expropriated homes being sold: Ex-owners face huge price tags to buy houses taken in failed Spadina Expressway bid". Toronto Star. p. B3.
^ Paperny, Anna Mehler (September 13, 2010). "Rossi pledges to tunnel Allen under downtown if elected". The Globe and Mail. Toronto. Retrieved September 13, 2010.
^ Osbaldeston 2008, pp. 144–149.
^ Moore, Michael (September 10, 1976). "Nearby Residents Worry About Traffic, But Motorists Love Spadina Extension". The Globe and Mail. Toronto, Ontario. p. 5.
^ Toronto Expressways – Transfer Points. February 2004. – p. 8.
^ "City of Toronto". Retrieved September 27, 2014.
^ "Toronto Neighbourhood Map". The Toronto Star. Retrieved April 30, 2010.
Colton, Timothy J. (1980). Big Daddy. Toronto, Ontario: University of Toronto Press. ISBN 0-8020-2393-2.
Nowlan, David; Nowlan, Nadine (1970). The Bad Trip: The Untold Story of the Spadina Expressway. Toronto, Ontario: new press/ House of Anansi. ISBN 0-88770-005-5.
Osbaldeston, Mark (2008). Unbuilt Toronto: A history of the city that might have been. Toronto, Ontario: Dundurn Press. ISBN 978-1-55002-835-5.
Rose, Albert (1972). Governing Metropolitan Toronto: A Social and Political Analysis 1953–1971. Berkeley and Los Angeles, California: University of California Press. ISBN 0-520-02041-3.
Sewell, John (1993). The Shape of the City: Toronto struggles with modern planning. Toronto, Ontario: University of Toronto Press. ISBN 0-8020-7409-X.
Toronto Planning Board, (1970). Evaluation of W. R. Allen Expressway. Toronto, Ontario: The Board.


To launch the first SimCity in almost a decade, Draftfcb San Francisco gave the world an interactive sneak peak of the game in a live online event called "Let's All Be Mayor." For seven days, the world joined forces to decide the fate of a different city each day and watched the results of their collective decisions play out in real time.
Reflecting on the stats that have come in from the game, we had some great results. It's no wonder that on March 31 "Let's All Be Mayor" was named an FWA (Favourite Website Awards)Site of the Day Award winner.
The FWA is an industry-recognized internet awards program with over 170 million site visits as of 2012. It is the most visited website award program in the history of the internet and is dedicated to recognizing amazing digital work.
Draftfcb San Francisco is very proud to have been part of this amazing digital campaign! If you didn't get a chance to play, take a minute to watch the recap videos on thesite.


The OHL All-Stars took a surprising outburst in the third period to seal the first win, and an all-around team effort in Game 2 to complete the sweep.
The best and brightest of the Ontario Hockey League took on the Russians in the ADT Canada-Russia challenge this week, and the OHL came away with two hard-fought wins.
The Russians began their trip across North America -- a six-game tour against all-star teams from each of the three CHL-member leagues -- in Chicoutimi and Gatineau, Quebec, and stole three of a possible four points from the Quebec Major Junior Hockey League all-stars.
The OHL All-Stars proved to be less hospitable. But it took a surprising outburst in the third period to seal the first win, and an all-around team effort in Game 2 to complete the sweep.
The Russians roared out with two goals in the first 10 minutes of Game 1, before Wayne Simmons of the Owen Sound Attack and Josh Bailey of the Windsor Spitfires responded for the OHL. The second period was also an even affair, with Kingston Frontenac Cory Emmerton, a Detroit Red Wing prospect, lighting the lamp for the OHL.
Then, in the third period with the score tied 3-3, the OHL exploded for two goals in 27 seconds, and the Russians were never able to regain control of the game. Oshawa Generals forward Dale Mitchell was credited with the game-winning goal while Nick Spaling of the Kitchener Rangers added the insurance goal.
Oshawa Generals forward John Tavares had three assists in the first game, and finished with a goal in the second game for a total of four points. He was named the OHL's Most Valuable Player in Game 1.
In the second game, it was Florida Panthers prospect and the OHL's fourth-leading scorer, Shawn Matthias, who led the team with a goal and an assist as the OHL defeated the Russians, 4-2.
Now, the Russians travel to Cranbrook B.C., for a game against the Western Hockey League All-Stars on Nov. 28 before wrapping up the 2007 ADT Canada-Russia challenge in Medicine Hat, Alberta, on Nov. 29.
Around the OHL -- Goaltender Jeremy Smith was named the Boston Pizza Player of the Week for the period ending Nov. 25, as he posted one shutout, a 1.33 GAA and a .964 save percentage in wins against Saginaw, Sault Ste. Marie and Sudbury. Smith currently sits third in the league in GAA with a 2.50 mark in 15 starts, and leads the OHL in save percentage with an impressive .933 percentage. … Kitchener Rangers forward Justin Azevedo had a five-point night this week, in a 6-3 victory against Sarnia. Azevedo scored three goals and added two assists in the game, and then followed that performance up with a goal and an assist in an 8-4 defeat of the Erie Otters the following evening. Azevedo's four-goal, three-assist week pushed him up the OHL scoring race to second overall, behind only John Tavares. Azevedo's total of 37 assists is tops in the OHL. … Brett MacLean, a second-round pick of the Phoenix Coyotes in 2007, scored twice Friday in a 6-5 shootout loss to the London Knights, and then added another goal Sunday in a 3-1 defeat of the Brampton Battalion. MacLean has now scored at least one goal in each of his last 11 games, and has averaged two goals per game during that stretch, posting a staggering 22 goals and eight assists for 30 points in only 11 games.
Chicago Steel forward Andy Miele currently sits tied for the league-lead in USHL scoring, due to an itchy trigger finger that has seen him score 18 goals in his first 17 games of 2007-08.
With a seven-point outburst in three games this week, he now sits with 26 points through 17 games, tied with John Kemp of the Indiana Ice, who has five goals and 21 assists through his first 16 games.
Miele began the week with a goal and an assists in an 8-3 win against Green Bay. He skated to a plus-2 rating and recorded four shots, notching a point in his third consecutive game.
He topped that performance in grand style Friday night, scoring four goals on four shots in a 10-3 defeat of the Indiana Ice. In addition to an assist (for five points on the evening), and a plus-3 rating, Miele also recorded 12 PIMs in the contest.
Miele will skate for the Miami (Ohio) RedHawks of the CCHA next season. The RedHawks are currently ranked No. 1 in the nation.
Around the USHL -- How do you feel if you're Andy Miele? Miele scores four goals in a game and records seven points as his team wins two of three games, and he doesn't get the RBK Offensive Player of the Week honors. That title went to teammate Drew Leblanc, who also scored five goals in the three games but added four assists for a total of nine points. He was plus-4 for the week. … With 19 goals scored in three games, you expect the Chicago Steel forwards to enjoy themselves, but a defenseman got into the act as well. Will Weber earned the CCM Defensive Player of the Week honors with a goal and two assists, a plus-5 rating and seven shots in the three games. … Sioux Falls goaltender Eric Hartzell won all three of his starts this week, knocking off the top two teams in the West Division. He posted a 3.00 GAA and a .917 save percentage in wins against Omaha, Sioux City, and Des Moines.
Halifax Mooseheads forward Tomas Knotek and Quebec Remparts forward Mikael Stefanovich may be rookies in name, but the two European-born skaters are playing like seasoned veterans in the QMJHL this season.
Knotek, of Kladno, Czech Republic, has 11 goals and 34 points in 28 games, pacing the 2007-08 QMJHL rookie class by four points over Stefanovich, a Minsk, Belarus native who has 15 goals and 30 points on the season.
Both Stefanovich and Knotek are highly ranked skaters in the recent CSS QMJHL preliminary rankings, with Stefanovich sitting atop the draft-eligible skaters and Knotek sitting in the top 20.
Around the QMJHL --Gatineau Olympiques forward Matthew Pistilli earned the TELUS Offensive Player of the Week for the period ending Nov. 25, as the winger scored a goal and added four assists in wins against Moncton and Val-d'Or. He now sits fourth in overall QMJHL scoring with 41 points in 30 games and his Olympiques sit atop the TELUS division standings. … Shawinigan defenseman Simon Lacroix was named the TELUS Defensive Player of the Week during that same stretch, as the seventh-round pick of the New York Islanders in 2007 scored twice and added two assists for four points in two games. Shawinigan won both of its games during the week, defeating Moncton and Acadie-Bathurst.
The Edmonton Oil Kings were the only team to mar the recent hot streak of the Lethbridge Hurricanes, so it seemed only fitting that they would also be the team to end the streak, when the Hurricanes finally ran out of luck.
The Oil Kings defeated the Hurricanes 6-3 on Nov. 21, marking the end of an incredible 14-game run in which Lethbridge was able to take 27 out of a possible 28 points, and remain undefeated in regulation for more than a month.
The previous game between the two teams, a 4-3 shootout win by the Oil Kings, was the single point Lethbridge failed to acquire during its run.
But all credit to the Hurricanes who suffered a shutout at the hands of the Vancouver Giants in the next game and still bounced back to gain a measure of revenge against the Oil Kings, defeating them 3-2 on Nov. 24 in front of Lethbridge's home fans.
The Hurricanes currently sit two points behind the Calgary Hitmen for first in the Central Division.
Around the WHL --The WHL is a tough place for rookies. Brandon Wheat Kings forward Brayden Schenn, with 24 points in 27 games this season, is the top-ranked rookie scorer in the league and sits 43rd in league scoring. He is the only rookie in the top 50 in WHL scoring, and leads the rookie class by two points over Tri-City Americans forward Johnny Lazo. By comparison, the top-ranked rookie scorer in the OHL, Patrick Maroon of the London Knights, has 34 points in 23 games, good for 21st in the league, and Tomas Knotek of the Mooseheads has 34 points in 28 games and sits tied for 13th in league scoring.


Drops from Garrosh Hellscream in Siege of Orgrimmar on normal difficulty. In a 10-man kill, dropped in addition to standard 2 pieces of loot.
Is this able to get on coinroll?
Just wanted to inform that i just got one in lfr on my coin roll.
does anyone know if this item is viable for item recovery? i accidentally sold it.


text:
GC NEWS
Greensboro College Presents Musical Theatre Performances March 28-29
Published by lex
GREENSBORO, N.C. – The Greensboro College Department of Theatre presents two musical-theatre performances, 7:30 p.m. Saturday-Sunday, March 28-29, in Hannah Brown Finch Memorial Chapel on campus.
Admission is free, and the public is invited. Free parking is available in the lot behind the college's Admissions Welcome Center at West Market Street and College Place.
The March 28 performance will feature Hannah Hjerth performing selections from "Trousers" and "Phantom of the Opera" and Brittany Corne performing selections from "Wicked" and "Dear World."
The March 29 performance will feature Kailey Rey performing her musical-theatre concentration senior recital. It also will feature Lindsey Mead performing her junior recital, including selections from "Grey Gardens" and "I'm Getting My Act Together and Taking It on the Road."
The Greensboro College Department of Theatre seeks to provide a strong foundation in theatre while allowing the student to emphasize in a particular area of theatre.
The coursework is integrated with the production work to provide a better understanding of the many facets of the theatre. Required participation on stage or backstage on all theatre productions allows the techniques and theories that are examined in the classroom and the laboratory to be tested in a performance setting.
Depending upon their emphasis of choice, upon completion of the degree, students will be prepared to enter graduate school or a theatre conservatory, intern full-time with a theatre organization, teach theatre or seek careers in the profession.
For more information, visit https://www.greensboro.edu/academics/school-of-the-arts/department-of-theatre/ or contact department chair David Schram at 336-272-7102, ext. 5243, or schramd@greensboro.edu.
Greensboro College provides a liberal arts education grounded in the traditions of the United Methodist Church and fosters the intellectual, social, and, spiritual development of all students while supporting their individual needs.
Founded in 1838 and located in downtown Greensboro, the college enrolls about 1,000 students from 29 states and territories, the District of Columbia, and seven foreign countries in its undergraduate liberal-arts program and six master's degree programs. In addition to rigorous academics and a well-supported Honors program, the school features a 17-sport NCAA Division III athletic program and dozens of service and recreational opportunities. Learn more at greensboro.edu.
Think critically. Act justly. Live faithfully.
Lex Alexander, Director of Communications
lex.alexander@greensboro.edu
www.greensboro.edu
Facebook: www.facebook.com/GreensboroCollege
Twitter: @GCPride
Instagram: www.instagram.com/greensborocollege
Tags: musical theatre, Theatre
Categorized in: News
This post was written by lex

redpajama_set_name: RedPajamaCommonCrawl