text:
\section{Introduction}\label{sec.intro}
Change point and data segmentation procedures have a long tradition in statistics. So called a-posteriori or offline procedures deal with the testing and segmentation of completely observed data, see e.g. \cite{aue2013} and \cite{horvath2014} for two recent survey articles mainly concerned with a-posteriori testing or \cite{fryzlewicz2014wild}, \cite{killick2012optimal}, \cite{rigaill2015pruned} and \cite{frick2014multiscale} to name but a few articles dealing with data segmentation.

On the other hand, data is collected more and more automatically with data arriving one by one, which requires a different statistical methodology designed to sequentially (online) make a decision after each new observation whether a break is likely to have occurred. In this setup an additional focus lies in the quick detection of changes after they have occurred. Examples include the monitoring of medical data of patients (e.g. \cite{fried2004online}), financial data (e.g. \cite{andreou2006monitoring}, \cite{aue2012sequential}) as well as deforestation monitoring (e.g. \cite{dutrieux2015monitoring}).

In sequential change point detection, there are different approaches: One approach aims at minimizing the mean detection delay while not causing too frequent alarms if no change occurs, see \cite{tarta} for a recent monograph.
In this paper, we follow a different approach closer related to classical testing theory, that was first proposed by
\cite{Chu}. By making use of a historic data set without changes their approach can control the asymptotic type-I-error while having power one in many situations.
This approach has been extended allowing for more general error sequences, multivariate observations but also different types of changes by several other authors including e.g.\
\cite{Hor}, \cite{aue2006change}, \cite{huvskova2005monitoring} or \cite{ciuperca2013two}. Additionally, \cite{chen2010modified}, \cite{fremdt2015page} and \cite{kirch2018modified} have proposed modified versions of the original monitoring scheme focusing more on recent monitoring observations to make a decision. These procedures outperform the original one for later changes while giving comparable results for early changes.

In this paper we present a unified theory based on $U$-statistics for different monitoring schemes. This is different from the unified theory based on estimating functions as proposed by
\cite{Est} and further considered in \cite{kirch2018modified}. In particular, the proposed methodology includes a robust Wilcoxon monitoring, thus adapting the a-posteriori methodology proposed by \cite{Dehl}.

\subsection{Change point problem and monitoring statistics}\label{sec.model}
Following  \cite{Chu}  we assume the existence of a historic data set $X_1,\ldots, X_m$ without a change. These historic observations can be used to estimate unknown parameters consistently. Asymptotic results are then obtained by letting the length of the historic data set increase to infinity. Subsequent to the historic observations we start monitoring new incoming data by testing for a structural break after each new observation $X_{m+k},k\geq 1,$ where $k$ denotes the monitoring time, by means of a monitoring statistic $\Psi(m,k)$ not depending on future data.
We consider the following general change model
\begin{equation}\label{meanmodel}
X_{i,m}=1_{\{1\leq i\leq k^*+m\}}Y_i+ 1_{\{i>k^*+m\}}Z_{i,m},\quad i\geq 1,
\end{equation}
where $\{Y_i\}_{i\in\mathbb{Z}}$ and $\{Z_{i,m}\}_{i\in\mathbb{Z}}$ are suitable stationary time series. The distribution of the time series after the change and thus the change itself is allowed to depend on $m$. The null hypothesis then corresponds to $k^
*=\infty$. Our examples focus on the special case of a classical mean change model with $Z_{i,m}=Y_i+d_m$, i.e.
\begin{equation}\label{meanex}
X_{i,m}=Y_i+1_{\{i>k^*+m\}}d_m,\quad d_m\neq 0,
\end{equation}
where the change in the mean $d_m$ is allowed to depend on m.\\

Because the monitoring continues as long as no alarm is given, the monitoring horizon is potentially infinite. By introducing a weight function $w(m,k)$ as e.g.\ given in \eqref{exw} below it is still possible to control the asymptotic type-I-error. More precisely, an alarm is given as soon as $$w(m,k)\left|\Psi(m,k)\right|>c_{\alpha},$$ where the critical value $c_{\alpha}$ is chosen such that the testing procedure holds the level $\alpha$ asymptotically. Equivalently (for $w(m,k)\neq 0$) an alarm is given as soon as the absolute monitoring statistic $\left|\Psi(m,k)\right|$ exceeds the critical curve given by $\frac{c_{\alpha}}{w(m,k)}$.

As long as the monitoring statistic does not exceed the critical curve, we
 continue monitoring, such that the stopping time $\tau_m$ is given by
 \begin{align*
				\tau_m=\begin{cases}
				\inf\{k\geq 1:w(m,k)\left|\Psi(m,k)\right|>c_{\alpha}\},\\
				\infty,\quad \mbox{if } w(m,k)\left|\Psi(m,k)\right|\leq c_{\alpha}\mbox{ for all k}.
				\end{cases}
				\end{align*}
				This setup allows to control the type-I-error as in classical statistics by choosing the critical value $c_{\alpha}$ for a given level $\alpha$ such that under the null hypothesis of no change
				$$
				\lim_{m\rightarrow\infty}P_{H_0}(\tau_m<\infty)=P_{H_0}\left(\sup_{k\geq 1}w(m,k)\left|\Psi(m,k)\right|> c_{\alpha}\right)=\alpha.
			$$
	Furthermore, it will turn out that typically under weak assumptions on the alternative such a sequential test also has asymptotic power one,				$$
				\lim_{m\rightarrow\infty}P_{H_1}(\tau_m<\infty)=1.
				$$

At each time point the question effectively comes down to a two-sample problem testing whether the distribution from the historic data set is still valid. For such a problem $U$-statistics have favorable properties, see e.g.\ \cite{ustat} for a recent monograph.  Therefore, we consider monitoring schemes that are based on the following $U$-statistic:
\begin{align}\label{detstat}
\Gamma(m,k)=\frac{1}{m}\sum_{i=1}^m\sum_{j=m+1}^{m+k}(h(X_i,X_j)-\theta),
\end{align}
where the kernel $h:\mathbb{R}^2\rightarrow\mathbb{R}$ is a measurable function and $\theta=\operatorname{E}(h(Y,Y_1))$ with $Y\stackrel{D}{=}Y_1$ being an independent copy of $Y_1$ (also independent of $\{Z_{i,m}\}$). The easiest example for a corresponding monitoring statistic is $\Psi(m,k)=\Gamma(m,k)$. The centering parameter $\theta$ is the expectation under the null hypothesis for independent data, while it
still approximates $\operatorname{E}(h(Y_i,Y_j))$ with increasing lag $j-i$ under appropriate weak dependency assumptions.
On the other hand, under alternatives the expectation (after the change) is given by $\operatorname{E}(h(Y,Z_{i,m}))$ so that changes can only be detected if $\operatorname{E}(h(Y,Z_{i,m}))\neq\theta$. The actual magnitude of the change is then given by
\begin{align}\label{eq_Delta}
	\Delta_m=\operatorname{E}(h(Y,Z_{i,m}))- \theta.
\end{align}
If we allow for local changes with $\Delta_m\to 0$, Theorem~\ref{TheoremH1w2} shows that changes are detectable with asymptotic power one if
\begin{align}\label{changedecay}
	\sqrt{m}|\Delta_m|\to\infty.
\end{align}

\subsection{Examples}\label{sec.ex.kernels}
In this section, we give three examples of suitable kernels, the first two will be considered in detail in the rest of the paper.
\subsubsection{Difference-of-means (DOM) statistic}\label{sec_dom}
The first papers on sequential change point statistics in the setup of \cite{Chu} all focus on the following difference-of-means (DOM) statistic

\begin{align}\label{cusumstat}
\Gamma_{D}(m,k)=\sum_{j=m+1}^{m+k} \left(\overline{X}_m-X_j\right)
=\frac{1}{m}\sum_{i=1}^m\sum_{j=m+1}^{m+k}\left(X_i-X_j\right).\end{align}
The second representation shows that the DOM statistic is a special case of the above $U$-statistics with the DOM kernel
\begin{align}\label{kernelC}
h_{D}(x_1,x_2)=x_1-x_2,
\end{align}
such that $\theta^{D}=\operatorname{E} h_{D}(Y,Y_1)=0$.
Under the alternative, we obtain
\begin{align}\label{delta.cusum}
\Delta^{D}_m=\operatorname{E} h_{D}(Y,Y_1+d_m)-\theta^{D}=-d_m\neq 0.
\end{align}
Consequently, \eqref{changedecay} is satisfied iff $\sqrt{m}|d_m|\rightarrow\infty$ as has already been pointed out by several authors.

\subsubsection{Wilcoxon statistic}\label{sec_wil}
In the classical two-sample situation, the Wilcoxon  kernel function
\begin{align}\label{kernelW}
h_W(x_1,x_2)=1_{\{x_1<x_2\}}
\end{align}
is frequently used. In this case, if $Y_1$ is continuous with density $f_Y$ and distribution function $F_Y$, it holds
\begin{align*
\theta^W=&\operatorname{E} h_W(Y,Y_1)=P(Y<Y_1)
=\operatorname{E}(F_Y(Y))=\frac{1}{2}.
\end{align*}
Consequently, the Wilcoxon statistic is given by
$$\Gamma_W(m,k)=\frac{1}{m}\sum_{i=1}^m\sum_{j=m+1}^{m+k}\left(1_{\{X_i<X_j\}}-1/2\right).$$
Under the alternative, it holds for $d_m>0$
\begin{align}
	\Delta_m^{W}&=\operatorname{E} h_W(Y,Y_1+d_m)-\theta^W
=P(Y<Y_1)+P(Y_1\leq Y<Y_1+d_m)-\theta^W\notag\\
&=P(Y_1\leq Y<Y_1+d_m)=\int_{-\infty}^{\infty}f_Y(z)\left(F_Y(z+d_m)-F_Y(z)\right)dz>0,
\label{eq_Delta_Wil}\end{align}
because $Y$ is a continuous random variable.
An analogous result is obtained for $d_m<0$.

\subsubsection{Further examples}
As pointed out in \cite{gombay} the kernel $h(x_1,x_2)=\frac 12(x_1+x_2)$ can be used for the detection of a change from a distribution $F$ that is symmetric around zero to a distribution $G$ that is not symmetric around zero with $Y_i\sim F$ and $Z_i\sim G$. This includes location shifts away from zero but also many shifts in shape.
The expected values of the kernel function under the null hypothesis and under the alternative are given by $\theta=\operatorname{E} \left(\frac 12(Y+Y_1)\right)=0$ and $\theta^*=\operatorname{E} \left(\frac 12(Y+Z_1)\right)=\operatorname{E}(Z_1).$\\

The procedures proposed in this paper require $\theta$ to be known and are thus restricted to kernel functions whose expected value under the null hypothesis does not depend on the (unknown) distribution of $Y_1$. Hence, for certain change point problems it might be useful to consider differences of one-sample $U$-statistics instead which is, however, outside the scope of this paper.

\subsection{Monitoring schemes}\label{monscheme}\label{sec.scheme}
The monitoring statistic as given in
(\ref{detstat})  with the DOM kernel is the one originally proposed by \cite{Chu}.
Because this monitoring scheme uses all observations of the monitoring period for the comparison with the historic data set, it uses many observations still following the null distribution if changes occur rather late -- resulting in smaller power and longer detection delay.
Therefore, the following adaptations ($\Psi_{2,3}$) of the monitoring scheme have been proposed in the literature focusing more on the recent monitoring observations for the comparison:
 $$
\begin{array}{ll}
\Psi_1(m,k):=\Gamma(m,k)=\frac{1}{m}\sum_{i=1}^m\sum_{j=m+1}^{m+k}(h(X_i,X_j)-\theta)&\mbox{CUSUM},\\
&\\
\Psi_2(m,k):=\Gamma(m,k)-\Gamma(m,\lfloor mb\rfloor)=\frac{1}{m}\sum_{i=1}^m\sum_{j=m+\lfloor kb\rfloor+1}^{m+k}(h(X_i,X_j)-\theta)&\mbox{mMOSUM},\\
&\\
\Psi_3(m,k):=\sup_{0\leq l\leq k}\left|\Gamma(m,k)-\Gamma(m,l)\right|&\\
\phantom{\Gamma_3(m,k)}\,\,=\sup_{0\leq l\leq k}\left|\frac{1}{m}\sum_{i=1}^m\sum_{j=m+l+1}^{m+k}(h(X_i,X_j)-\theta)\right|&\mbox{Page-CUSUM}.
\end{array}
$$
The fixed tuning parameter $b\in (0,1)$ in the modified MOSUM (mMOSUM) determines the percentage of the earlier observations that are discarded, whereas the Page-CUSUM does not require an a priori choice of such a parameter.\\
The modified MOSUM (mMOSUM) has been proposed in \cite{chen2010modified} and the Page-CUSUM in \cite{fremdt2015page}, both for the linear model. \cite{horvath2008performance} and  \cite{aue2012reaction} consider a standard MOSUM statistic for the mean change model, however, in a comparison of all of those statistics \cite{kirch2018modified} conclude that the MOSUM procedure has noticeable power problems although its detection delay is very small for those changes that are detected.
Because we are interested in a reliable testing procedure in the sense of having a large power, we concentrate on the above three monitoring schemes in this paper.

\vspace{2mm}

The paper is organized as follows: In Section \ref{sec.as}, we derive the asymptotic theory for the above monitoring schemes. The limit distribution of the test statistics under the null hypothesis provided in Section \ref{sec.asH0} allws to control the asymptotic type-I-error. In Section \ref{sec.asH1}, we show that the proposed procedures have asymptotic power one. The asymptotic results are based on  general assumptions which are examined for specific kernels and dependency structures in Section \ref{sec.exass}. The finite sample behavior of the sequential procedures is investigated in a simulation study in Section \ref{sec.sim} where we compare different kernels and monitoring schemes. A data example is provided in Section \ref{sec.data}. Proofs are postponed to an Appendix.

\section{Asymptotics}\label{sec.as}
We provide the theory for the general $U$-statistics stated in Section \ref{sec.scheme}. First, in Section \ref{sec.asH0}, we derive the asymptotics under the null hypothesis in order to control the type-I error. Consistency of the procedures is shown in Section \ref{sec.asH1}. The assumptions on the dependency structure of the observed time series depend on the kernel function and are rather abstract. In Section~\ref{sec.exass} we derive them explicitly for both  independent observations as well as  functionals of mixing processes.

\subsection{Asymptotics under the null hypothesis}\label{sec.asH0}
In order to state the assumptions we need to introduce Hoeffding's decomposition which is a well-known tool when dealing with $U$-statistics. As proposed by \cite{Hoeff} the kernel function is decomposed under the null hypothesis as follows:
		\begin{align}\label{Hdec}
		h(x_1,x_2)=\theta+h_1(x_1)+h_2(x_2)+r(x_1,x_2)\end{align}
				with
				\begin{align*}
				&\theta=\operatorname{E}(h(X_1,X_2)),\qquad
				h_1(x_1)=\operatorname{E}(h(x_1,X_2))-\theta,\qquad				h_2(x_2)=\operatorname{E}(h(X_1,x_2))-\theta,\notag\\
			&	r(x_1,x_2)=h(x_1,x_2)-h_1(x_1)-h_2(x_2)-\theta,
				\end{align*}

				where $X_1$ and $X_2$ are independent random variables with the same distribution.
				In particular,
				\begin{align}\label{hcenter}
				\operatorname{E}\left(h_1(X_1)\right)=\operatorname{E}\left(h_2(X_2)\right)=0
				\end{align}
				For the DOM kernel as in Section~\ref{sec_dom} we obtain
\begin{align}\label{eq_h_dom}
&h^D_1(x_1)=x_1-E(X_2),\quad
h^D_2(x_2)=\operatorname{E}(X_1)-x_2=-h^D_1(x_2),\quad r^D(x_1,x_2)=0.
\end{align}

For the Wilcoxon kernel for continuous random variables $X_1$ and $X_2$ with distribution function $F$ is given by
\begin{align}\label{eq_h_wil}
&h^W_1(x_1)=\frac 12-F(x_1),\quad h^W_2(x_2)=-h^W_1(x_2),\quad r^W(x_1,x_2)=1_{\{x_1<x_2\}}+F(x_1)-F(x_2)-\frac 12.
\end{align}

We provide the asymptotic results for a general class of weight functions with the following regularity conditions that have also been used in \cite{Est} and \cite{Weber}:
\begin{assumption}\label{regw} Let the weight function satisfy
$ $
\begin{itemize}
\item[(i)] $w(m,k)=m^{-1/2}\tilde{w}(m,k),$ where
$\tilde{w}(m,k)=\rho\left(\frac km\right)$ for $k>l_m$ with $\frac{l_m}{m}\rightarrow 0$ and $\tilde{w}(m,k)=0$ for $k\leq l_m.$ The function $\rho:(0,\infty)\rightarrow\mathbb{R}^+$ is positive and continuous.
\end{itemize}
The following two limits exist and it holds:
\begin{itemize}
\item[(ii)] $\lim_{t\rightarrow 0}t^{\gamma}\rho(t)<\infty$ for some $0\leq\gamma<\frac 12.$
\item[(iii)] $\lim_{t\rightarrow\infty}t\rho(t)<\infty.$
\end{itemize}
\end{assumption}

Part (i) allows to start the monitoring only after some observations have been collected. This is inmportant in order to avoid false alarms at the very beginning of the monitoring period. Such early false alarms often occur  as the first values of the monitoring statistic are quite volatile due to the small monitoring sample (but are asymptotically negligible).

The regularity conditions of the weight function at zero and infinity in  (ii) and (iii) are needed to control the asymptotic behavior of the monitoring statistic at 0 and inifinity.

The following choice  fulfills these assumptions and is often used in the literature as it leads to nice limit  distributions as indicated in Corollary \ref{simple1}:
\begin{align}\label{exw}
\rho(t)=\frac{1}{1+t}\left(\frac{1+t}{t}\right)^{\gamma},
\quad 0\leq\gamma<\frac 12.
\end{align}

Instead of using explicit assumptions on the underlying dependency structure of the observed time series we work with the following high-level regularity assumptions allowing for easy future extensions to different dependency assumptions. In Section~\ref{sec.exass} we show their validity for both  independent observations as well as  functionals of mixing processes.
\begin{assumption}\label{regass}
Let $\{Y_i\}_{i\in\mathbb{Z}}$ be a stationary time series that fulfills the following assumptions for a given kernel function $h$.
\begin{enumerate}
\item[(i)]
$\operatorname{E}\left(\left|\sum_{i=1}^m\sum_{j=k_1}^{k_2}r(Y_i,Y_j)\right|^2\right)\leq u(m)(k_2-k_1+1)\quad\mbox{for all } m+1\leq k_1\leq k_2$\\
with  $\frac{u(m)}{m^{2-2\gamma}}\log(m)^2\rightarrow 0$ and $\gamma$ as in Assumption \ref{regw}.\\
\item[(ii)] The following functional central limit theorem holds for any $T>0$
$$\left\{\frac{1}{\sqrt{m}}\sum_{i=1}^{[mt]}\left(h_1(Y_i),h_2(Y_i)\right):0< t\leq T\right\}\stackrel{D}{\rightarrow}\left\{\left(\tilde{W}_1(t),\tilde{W}_2(t)\right):0< t\leq T\right\},$$
where $\left\{\left(\tilde{W}_1(t),\tilde{W}_2(t)\right):0< t\leq T\right\}$ is a bivariate Wiener process with mean zero and covariance matrix
$$
\Sigma=\begin{pmatrix}
\sigma_1^2 & \rho\notag\\
\rho &\sigma_2^2
\end{pmatrix}
$$
with \begin{align}\label{defsig}
\sigma_1^2=\sum_{h\in\mathbb{Z}}\operatorname{Cov}(h_1(Y_0),h_1(Y_h)),\quad \sigma_2^2=\sum_{h\in\mathbb{Z}}\operatorname{Cov}(h_2(Y_0),h_2(Y_h)).
\end{align}
\item[(iii)] For all $0\leq\alpha<\frac{1}{2}$ the following H\'ajek-R\'enyi-type inequality holds
$$
\sup_{1\leq k\leq m}\frac{1}{m^{\frac{1}{2}-\alpha}k^{\alpha}}\left|\sum_{j=1}^{k}h_2(Y_j)\right|=O_P\left(1\right)\quad\mbox{ as }m\rightarrow\infty.
$$
\item[(iv)]The following H\'ajek-R\'enyi-type inequality holds uniformly in $m$ for any sequence $k_m>0$
$$
\sup_{k\geq k_m}\frac{1}{k}\left|\sum_{j=1}^{k}h_2(Y_j)\right|=O_P\left(\frac{1}{\sqrt{k_m}}\right)\quad\mbox{as }k_m\rightarrow\infty.
$$
\end{enumerate}
\end{assumption}
Assumption~\ref{regass} (i) is required to prove the asymptotic negligibility of the remainder term, while (ii) is a classic functional central limit theorem providing the limit distribution. Note that  $\sigma_j^2$, $j=1,2$, are the corresponding long-run variances which coincide with the variances only in the independent case. In
this case, we obtain
\begin{align}\label{eq_var_dom}
\sigma_{1,D}^2=\sigma_{2,D}^2=\operatorname{Var}(Y_1)
\end{align}
with \eqref{eq_h_dom} for the DOM kernel and
\begin{align}\label{eq_var_wil}
\sigma_{1,W}^2=\sigma_{2,W}^2=\operatorname{Var}\left(F(Y_1)\right)=\frac{1}{12}
\end{align}
with \eqref{eq_h_wil} for the Wilcoxon kernel if $Y_1$ is continuous with distribution function $F$.
This shows in particular, that in the independent case the Wilcoxon statistic does not require an estimator of the variance. Unfortunately, this is no longer true in the time series case as the long-run variance of the Wilcoxon kernel is not distribution-free for continuous random variables and thus requires an estimator.

The H\'ajek-R\'enyi-type inequalities in (iii) and (iv) are also standard tools in this context that hold for many dependency concepts (see e.g. Section B.1 in \cite{kdiss}), they are used in combination with Assumption~\ref{regw} (ii) -- (iii) to control the (asymptotic) behavior of the statistic at the beginning and end of the monitoring period.

\begin{theorem}\label{as.H0}
Let the regularity conditions given in Assumption \ref{regw} and \ref{regass} be fulfilled. Then, it holds under $H_0$, as $m\rightarrow\infty$,
\begin{itemize}
\item[(i)] $\sup_{k\geq 1}w(m,k)\left|\Psi_1(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{t>0}\rho(t)\left|\sigma_2W_2(t)+t\sigma_1W_1(1)\right|$,
\item[(ii)] $\sup_{k\geq 1}w(m,k)\left|\Psi_2(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{t>0}\rho(t)\left|\sigma_2(W_2(t)-W_2(tb))+t(1-b)\sigma_1W_1(1)\right|$,
\item[(iii)] $
\sup_{k\geq 1}w(m,k)\left|\Psi_3(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{t>0}\rho(t)\sup_{0<s\leq t}\left|\sigma_2(W_2(t)-W_2(s))+(t-s)\sigma_1W_1(1)\right|$,
\end{itemize}
for two independent standard Wiener processes $\{W_1(t):t>0\}$ and $\{W_2(t):t>0\}$.
\end{theorem}
While the above limit distributions can already be used in combination with the Wilcoxon kernel and independent data, where the variance is completely known (see \eqref{eq_var_wil}), this is not the case for the CUSUM statistic or dependent data.

In such cases, the unknown parameters can be estimated consistently based on the historic data set, leading to the following pivotal limit distribution. Furthermore, this limit distribution can now be written as a supremum over the compact interval $[0,1]$ rather than a supremum over the positive real numbers.

\begin{corollary}\label{simple1}
Let $\{W(t):t>0\}$ be a standard Wiener processes.
\begin{itemize}
	\item[a)] If $\sigma_1=\sigma_2=:\sigma$ and $\hat{\sigma}_m\stackrel{P}{\rightarrow}\sigma$, then
\begin{itemize}
\item[(i)] $\frac{1}{\hat{\sigma}_m}\sup_{k\geq 1}w(m,k)\left|\Psi_1(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{0<t<1}\rho\left(\frac{t}{1-t}\right)\frac{|W(t)|}{1-t}$,
\item[(ii)] $\frac{1}{\hat{\sigma}_m}\sup_{k\geq 1}w(m,k)\left|\Psi_2(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{0<t<1}\rho\left(\frac{t}{1-t}\right)\left|\frac{W(t)}{1-t}-(1-t(1-b))\frac{W\left(\frac{tb}{1-t(1-b)}\right)}{1-t}\right|$,
\item[(iii)] $\frac{1}{\hat{\sigma}_m}\sup_{k\geq 1}w(m,k)\left|\Psi_3(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{0<t<1}\rho\left(\frac{t}{1-t}\right)\sup_{0<s\leq t}\left|\frac{W\left(t\right)}{1-t}-\frac{W\left(s\right)}{1-s}\right|$.
\end{itemize}
For a weight function as in \eqref{exw} these limit distributions further simplify as in this case \mbox{$\rho\left( \frac{t}{1-t} \right)=\frac{1}{t^{\gamma}}\,(1-t)$.}
\item[b)] For $\sigma_1\neq\sigma_2$ and $\hat{\sigma}_{1,m}\stackrel{P}{\rightarrow}\sigma_1$ as well as $\hat{\sigma}_{2,m}\stackrel{P}{\rightarrow}\sigma_{2}$ it holds
\begin{itemize}
\item[(i)] $\sup_{k\geq 1}\frac{\hat{\sigma}_{1,m}}{\sqrt{m}\left(\hat{\sigma}_{2,m}^2+\hat{\sigma}_{1,m}^2\frac{k}{m}\right)}\left|\Psi_1(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{0< t< 1}\left|W(t)\right|$,
\item[(ii)] $\sup_{k\geq 1}\frac{\hat{\sigma}_{1,m}}{\sqrt{m}\left(\hat{\sigma}_{2,m}^2+\hat{\sigma}_{1,m}^2\frac{k}{m}\right)}\left|\Psi_2(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{0<t<1}\left|W(t)-(1-t(1-b))W\left(\frac{tb}{1-t(1-b)}\right)\right|$,
\item[(iii)] $\sup_{k\geq 1}\frac{\hat{\sigma}_{1,m}}{\sqrt{m}\left(\hat{\sigma}_{2,m}^2+\hat{\sigma}_{1,m}^2\frac{k}{m}\right)}\left|\Psi_3(m,k)\right|\stackrel{\mathcal{D}}{\rightarrow}\sup_{0<t<1}\sup_{0<s\leq t}\left|W\left(t\right)-\frac{1-t}{1-s}W\left(s\right)\right|$.
\end{itemize}
\end{itemize}
\end{corollary}

For both the DOM- and the Wilcoxon kernel it holds $h_2(y)=-h_1(y)$ and thus  $\sigma_1=\sigma_2$.

\subsection{Consistency under alternatives}\label{sec.asH1}
In the following we show that the proposed procedures have asymptotic power one (for $m\to\infty$) under alternatives, i.e.\ the procedure will eventually stop under alternatives.

In the situation of \eqref{meanmodel} we get with $\theta_m^*:=\operatorname{E}(h(Y,Z_{i,m})=\Delta_m+\theta$ (see \eqref{eq_Delta}) for $k>k^*$
\begin{align*}
\Gamma(m,k)=&\frac{1}{m}\sum_{i=1}^m\sum_{j=m+1}^{m+k^*}(h(Y_i,Y_j)-\theta
+\frac{1}{m}\sum_{i=1}^m\sum_{j=m+k^*+1}^{m+k}(h(Y_i,Z_{j,m})-\theta_m^*)+(k-k^*)\Delta_m.
\end{align*}

The first summand is the same term as under the null hypothesis which has already been analyzed in the previous section. The second summand is a related two-sample $U$-statistic for which the two samples are no longer generated by the same distribution, while the third summand is the signal term.

In order to control the second term, consider the following version of Hoeffding's decomposition:
\begin{align}\label{hoeffh1}
h(y,z)=\theta_m^*+h^*_{1,m}(y)+h^*_{2,m}(z)+r^*_m(y,z)
\end{align}
where  we define for independent random variables $Y\stackrel{\mathcal{D}}{=}Y_1$ and $Z_m\stackrel{\mathcal{D}}{=}Z_{m,1}$
\begin{align*}
&h^*_{1,m}(y)=\operatorname{E} h(y,Z_m)-\theta_m^*, \quad\theta_m^*=\operatorname{E} h(Y,Z_m),\quad
h^*_{2,m}(z)=h_2(z)-\Delta_m,\quad \Delta_m=\theta_m^*-\theta,
\\
&r_m^*(y,z)=h(y,z)-h^*_{1,m}(y)-h^*_{2,m}(z)-\theta_m^*.
\end{align*}
 Again, in this decomposition, $h_{1,m}^*(Y)$, $h_{2,m}^*(Z_m)$ as well as $r_m^*(Y,Z_m)$ are centered random variables.

 For the DOM kernel in model \eqref{meanex} this results in
\begin{align*}
&h^{*D}_{1,m}(y
=y-\operatorname{E}(Y),\quad
h^{*D}_{2,m}(z)
\operatorname{E}(Y)-z+d_m,\quad
r^{*D}(y,z
=0.
\end{align*}
For the Wilcoxon kernel, model \eqref{meanex} and a continuous $Y$ with distribution function $F$  it holds with \eqref{eq_Delta_Wil}
\begin{align*}
&h_{1,m}^{*W}(y
=\frac 12-F(y-d_m)-\Delta_m^W,\quad
h_{2,m}^{*W}(z
=F(z)-\Delta_m^W+\frac 12,\quad\\
&r^{*W}(y,z
=1_{\{y<z\}}+F(y-d_m)-F(z)+\Delta_m^W-\frac 32
\end{align*}

In order to obtain asymptotic power one, we need to impose some additional assumptions on the weight function, which are again fulfilled for the weight function  in \eqref{exw}.
\begin{assumption}\label{altrho}
\begin{itemize}
\item[(i)] If  $\frac{k^*}{m}\rightarrow\infty$, assume that $\liminf_{t\rightarrow\infty}t\rho(t)>0.$
\item[(ii)] If $\frac{k^*}{m}=O(1),$ i.e. $\frac{k^*}{m}<\nu$ for all $m\geq 1$ for some $\nu>0$, assume that there exist $t_0>\nu, \epsilon>0$ such that $\rho(t)>0$ for all $t\in (t_0-\epsilon, t_0+\epsilon)$.
\item[(iii)] If $\frac{k^*}{m}=O(1),$ i.e. $\frac{k^*}{m}<\nu$ for all $m\geq 1$ for some $\nu>0$, assume that there exist $t_0>\nu, \epsilon>0$ such that $\rho(t)>0$ for all $t\in \left(\frac{t_0}{b}-\epsilon, \frac{t_0}{b}+\epsilon\right)$.
\end{itemize}
\end{assumption}

We assume that the time series before the change fulfills the regularity conditions under the null hypothesis as given in Assumption \ref{regass}.
For the time series after the change $\{Z_{i,m}\}$ the following weaker assumptions are required.
\begin{assumption}\label{regassH1}
Let $\{Y_i\}_{i\in\mathbb{Z}}$ and $\{Z_{i,m}\}_{i\in\mathbb{Z}}$ be stationary time series that fulfill the following assumptions for a given kernel function $h$.
\begin{enumerate}
\item[(i)]
$\sum_{i=1}^m\sum_{j=k_1}^{k_2}r_m^*(Y_i,Z_{j,m})=O_P( \sqrt{m\,(k_2-k_1+1)\,\max(m,k_2-k_1+1)})$ for all sequences $k_1(m)\le k_2(m)$.
\item[(ii)] $\frac{1}{\sqrt{m}}\sum_{i=1}^mh^*_{1,m}(Y_i)=O_p(1)$ as $m\rightarrow\infty$.
\item[(iii)] $\frac{1}{\sqrt{k_m}}\sum_{j=m+k^*+1}^{m+k^*+k_m}h^*_{2,m}(Z_{j,m})=O_p(1)$ as $k_m\rightarrow\infty$.
\end{enumerate}
\end{assumption}

The following theorem shows that the monitoring will eventually stop and the null hypothesis be rejected with asymptotic probability one under the alternative.
\begin{theorem}\label{TheoremH1w2}
	Let the regularity conditions given in Assumptions \ref{regw}, \ref{regass} (for $\{Y_i\}$) and \ref{regassH1}  be satisfied. Furthermore assume that $\sqrt{m}|\Delta_m|\rightarrow\infty$ and that Assumption~\ref{altrho} (i) holds.
	Then, it holds under the alternative $$\sup_{k\geq 1}w(m,k)\left|\Psi_j(m,k)\right|\stackrel{P}{\rightarrow}\infty$$ for $j=1,3$ if Assumption \ref{altrho} (ii) as well as for $j=2$ if Assumption~\ref{altrho} (iii)  is fulfilled.
\end{theorem}

\subsection{Regularity assumptions}\label{sec.exass}
In this section, we show that
Assumption~\ref{regass} is fulfilled in the i.i.d.\ case under certain moment assumptions as well as under the weak dependency concept that has also been used in \cite{Dehl}.

We first properly define the latter:
A stochastic process $\{Z_i:i\in\mathbb{Z}\}$ is called absolutely regular (see \cite[Definition 1.1]{borovkova2001limit}) if
\begin{align*}
\beta(k)&=\sup_{i\geq 1}\left\{\operatorname{E}\left(\sup_{A\in\mathcal{A}_{i+k}^{\infty}}|P(A|\mathcal{A}_{-\infty}^i)-P(A)|\right)\right\}\notag\\
&=\frac 12\sup_{i\geq 1}\left\{\sup\sum_{j=1}^J\sum_{l=1}^L|P(A_j\cap B_l)-P(A_j)P(B_l)| \right\}\rightarrow 0\quad (k\rightarrow\infty)
\end{align*}
where $\mathcal{A}_{i_1}^{i_2}=\sigma(Z_{i_1},Z_{i_1+1}\ldots, Z_{i_2})$ and the inner supremum in the second representation is taken over all finite $\mathcal{A}_{-\infty}^i-$ measurable partitions $(A_1,\ldots,A_J)$ and all finite $\mathcal{A}_{i+k}^{\infty}-$ measurable partitions $(B_1,\ldots,B_L)$, $J,L$ arbitrary. A sequence $\{Y_i:i\geq 1\}$ is called an r-approximating functional (see \cite[Definition 1.4.]{borovkova2001limit}) with approximating constants $\{a_k\}_{k\geq 0}$  of $\{Z_i:i\in\mathbb{Z}\}$ if
$$\operatorname{E}|Y_0-\operatorname{E}(Y_0|Z_{-k},\ldots,Z_k)|^r\leq a_k$$
with $a_k\rightarrow 0$ as $k\rightarrow\infty$. A kernel $h:\mathbb{R}^2\rightarrow\mathbb{R}$ is called 1-continuous, if there exists a function $\Phi:(0,\infty)\rightarrow (0,\infty)$ with $\Phi(\epsilon)\rightarrow 0$ as $\epsilon\rightarrow 0$ such that for all $\epsilon>0$
\begin{align*}
&\operatorname{E}\left(|h(X',Y)-h(X,Y)|1_{\{|X-X'|<\epsilon\}}\right)\leq \Phi(\epsilon)\notag\\
&\operatorname{E}\left(|h(X,Y')-h(X,Y)|1_{\{|Y-Y'|<\epsilon\}}\right)\leq \Phi(\epsilon)
\end{align*}
for all random variables $X,X',Y$ and $Y'$ having the same marginal distribution as $Y_1,$ and such that $X,Y$ are either independent or have joint distribution $P_{(Y_1,Y_k)}$ for some integer k (see Definition 2 in \cite{Dehl}).

\begin{lemma}\label{ass.ex}
\begin{itemize}
\item[a)] Let $Y_1,Y_2,\ldots$ be i.i.d.\ random variables
	and let $h:\mathbb{R}^2\rightarrow\mathbb{R}$ be a kernel with
\begin{align}\label{ass.sym}
Eh^2(Y_1,Y_2)<\infty,\quad
0<\operatorname{E}\left|h_1(Y_1)\right|^{\nu}<\infty,\quad
0<\operatorname{E}\left|h_2(Y_1)\right|^{\nu}<\infty
\end{align}
for some $\nu>2$. Then, Assumptions \ref{regass} are fulfilled.
\item[b)] Let $\{Y_i:i\geq 1\}$ be a 1-approximating functional with approximating constants $\{a_k\}_{k\geq 0}$ of an absolutely regular process with mixing coefficients $\{\beta(k)\}_{k\geq 0}$ and let $h(x,y)$ be a bounded 1-continuous kernel. Then, Assumptions \ref{regass} are fulfilled if
\begin{align}\label{coeff.cond}
\sum_{k\geq 1}k^2\left(\beta(k)+\sqrt{a_k}+\Phi(\sqrt{a_k})\right)<\infty.
\end{align}
\end{itemize}
\end{lemma}
The  Wilcoxon kernel is bounded and 1-continuous so that Lemma~\ref{ass.ex} b) applies.

The DOM kernel, on the other hand,  is clearly not bounded such that Lemma \ref{ass.ex} b) does not apply. However, for the DOM kernel, Assumptions~\ref{regass} can be shown for many weak dependency concepts: Note, in particular, that due to remainder term being exactly zero, Assumption~\ref{regass} (i) holds trivially for the DOM kernel. Because $h_1^D(Y_1)=Y_1-\operatorname{E}(Y_1)=-h_2^D(Y_1)$ (ii) comes down to a functional central limit theorem for $\{Y_i\}$, which has been derived under various assumptions on the dependency.
Similarly, the H\'ajek-R\'enyi-type inequalities in (iii) and (iv) (for $Y_i-\operatorname{E}(Y_i)$) have been shown under rather general assumptions. For example, as described below Assumption A.3 in \cite{Est}, all three assumptions follow from a strong invariance principle that has been derived for many processes (see e.g.\ Sections 2.4 in \cite{Auediss} for some examples).

\section{Simulation study and data analysis}
\subsection{Simulation study}\label{sec.sim}

In this section, we illustrate the finite sample performance of the sequential procedures comparing different monitoring schemes, different choices of $\gamma$ in \eqref{exw} as well as the DOM- and Wilcoxon kernel via their empirical size, power and stopping time. All empirical results are based on $10\, 000$ replications for each scenario.

We consider the mean change model as in (\ref{meanex}) with $d_m=0.5$, $m=100$ and independent innovations $\{Y_i\}$ with a standard normal distribution as well as a standardized $t_3$ distribution, where under the alternative we consider early $k^{*}=m^{\beta}$ with $\beta=0.5$, medium late $\beta=1$ as well as late changes $\beta=1.4$. The asymptotic critical values are obtained based on $50\,000$ realizations of the limit distributions in Corollary \ref{simple1} b) where the Wiener processes are approximated on a grid of $10\, 000$ points.

For the Wilcoxon kernel, the variance as in \eqref{eq_var_wil} does not depend on the distribution of $Y_1$, while for the DOM statistics $\operatorname{Var}(Y_1)$ (see \eqref{eq_var_dom}) is estimated by the empirical variance of the historic data set $\widehat{\sigma}_m^2=\frac{1}{m-1}\sum_{j=1}^m\left(X_{j,m}-\frac{1}{m}\sum_{l=1}^mX_{l,m}\right)^2$.

Following \cite{kirch2018modified} we start monitoring only after $a_m=\sqrt{m}=10$ in all settings in order to avoid false positives at the very beginning of the monitoring period. This does not affect the null asymptotics.

\begin{table}[tb]
\begin{minipage}{0.48\textwidth}
\textbf{Standard normal distribution:}
\vspace{-0.3cm}
\begin{center}
\begin{adjustbox}{max width=\textwidth}
\(
\begin{tabu}{l||c|c|c||c|c|c|}
\multicolumn{1}{c||}{\textbf{Monitoring}}&\multicolumn{3}{c||}{\mbox{\textbf{DOM kernel}}}&\multicolumn{3}{c|}{\mbox{\textbf{Wilcoxon kernel}}}\\
\multicolumn{1}{c||}{\cellcolor{white}\textbf{scheme}}&\boldsymbol{\gamma=0\,\,\,\,}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0\,\,\,\,}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}\\\hline
\mbox{\textbf{CUSUM}}&4.70&4.72&3.69&4.26&4.40&3.13\\\hline
\mbox{\textbf{Page-CUSUM}}&4.55&4.55&3.22&4.25&4.18&2.52\\\hline
\mbox{\textbf{mMOSUM}}&&&&&&\\
\multicolumn{1}{r||}{b=0.1}&4.62&4.83&3.61&4.35&4.48&2.91\\\hline
\multicolumn{1}{r||}{b=0.4}&4.95&5.08&3.07&4.84&4.31&2.25\\\hline
\multicolumn{1}{r||}{b=0.9}&4.90&5.28&3.94&2.09&0.86&0.03\\\hline
\end{tabu}
\)
\end{adjustbox}\\
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\textbf{Standardized $t_3$-distribution:}
\vspace{-0.3cm}
\begin{center}
\begin{adjustbox}{max width=\textwidth}
\(
\begin{tabu}{l||c|c|c||c|c|c|}
\multicolumn{1}{c||}{\textbf{Monitoring}}&\multicolumn{3}{c||}{\mbox{\textbf{DOM kernel}}}&\multicolumn{3}{c|}{\mbox{\textbf{Wilcoxon kernel}}}\\
\multicolumn{1}{c||}{\cellcolor{white}\textbf{scheme}}&\boldsymbol{\gamma=0\,\,\,\,}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0\,\,\,\,}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}\\\hline
\mbox{\textbf{CUSUM}}& 5.56& 6.87& 6.93& 4.39& 4.36& 3.12\\\hline
\mbox{\textbf{Page-CUSUM}}& 5.79& 7.12& 6.50& 4.27& 4.09& 2.37\\\hline
\mbox{\textbf{mMOSUM}}&&&&&&\\
  \multicolumn{1}{r||}{b=0.1}&6.24& 7.71& 6.86& 4.51& 4.34& 2.78\\\hline
 \multicolumn{1}{r||}{b=0.4}& 8.64&10.02& 8.41& 4.46& 3.80& 1.94\\\hline
\multicolumn{1}{r||}{b=0.9}&29.26&31.53&24.86& 2.30& 0.98& 0.03\\\hline
\end{tabu}
\)
\end{adjustbox}
\end{center}
\end{minipage}
\caption{Empirical size (in $\%$) for a nominal level of $\alpha=5\%$ for independent innovations.}
\label{empsize}

\end{table}

Table \ref{empsize} shows the empirical size. For the standard normal distribution the procedures hold the nominal level and become more and more conservative with increasing $\gamma$. This is still true for the $t_3$-distribution when using the Wilcoxon kernel whereas the DOM kernel in combination with the mMOSUM and $b=0.9$ leads to high rejection rates, which occur mainly at the beginning of the monitoring period where only few summands are included in the monitoring statistics.
While for the normal distribution, this effect can be controlled by an appropriate choice of $a_m$, for the $t_3$-distribution and large values of $b$ this is not sufficient. Hence, one should be careful when using large $b$ for heavy tailed observations.

\begin{table}[htb]
\textbf{DOM kernel:}
\vspace{-0.3cm}
\begin{center}
\begin{adjustbox}{max width=0.9\textwidth}
\(
\begin{tabu}{l||c|c|c||c|c|c||c|c|c|}
&\multicolumn{3}{c||}{\boldsymbol{\beta=0.25,\quad k^{*}=3}}&\multicolumn{3}{c||}{\boldsymbol{\beta=1,\quad k^{*}=100}}&\multicolumn{3}{c|}{\boldsymbol{\beta=1.4,\quad k^{*}=630}}\\
\multicolumn{1}{c||}{\cellcolor{white}\textbf{Monitoring scheme}}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}\\\hline
\mbox{\textbf{CUSUM}}&99.75&99.69&99.18&99.27&99.02&98.08&87.74&84.78&76.85\\\hline
\mbox{\textbf{Page-CUSUM}}&99.74&99.68&99.21&99.45&99.34&98.52&94.99&92.65&84.28\\\hline
\mbox{\textbf{mMOSUM}}&&&&&&&&&\\
\multicolumn{1}{r||}{b=0.1}&99.75&99.60&99.05&99.61&99.43&98.66&93.20&90.52&82.11\\\hline
\multicolumn{1}{r||}{b=0.4}&99.59&99.25&97.54&99.53&99.07&96.84&99.41&98.39&93.61\\\hline
\multicolumn{1}{r||}{b=0.9}&89.42&71.36&36.51&88.43&64.74&19.28&72.85&33.80& 6.00\\\hline
\end{tabu}
\)
\end{adjustbox}
\end{center}
$ $\\
\textbf{Wilcoxon kernel:}
\vspace{-0.3cm}
\begin{center}
\begin{adjustbox}{max width=0.9\textwidth}
\(
\begin{tabu}{l||c|c|c||c|c|c||c|c|c|}
&\multicolumn{3}{c||}{\boldsymbol{\beta=0.25,\quad k^{*}=3}}&\multicolumn{3}{c||}{\boldsymbol{\beta=1,\quad k^{*}=100}}&\multicolumn{3}{c|}{\boldsymbol{\beta=1.4,\quad k^{*}=630}}\\
\multicolumn{1}{c||}{\cellcolor{white}\textbf{Monitoring scheme}}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}\\\hline
\mbox{\textbf{CUSUM}}&99.68&99.59&99.00&99.10&98.84&97.79&86.52&83.59&74.78\\\hline
\mbox{\textbf{Page-CUSUM}}&99.66&99.55&99.02&99.39&99.13&98.20&94.68&91.65&82.25\\\hline
\mbox{\textbf{mMOSUM}}&&&&&&&&&\\
\multicolumn{1}{r||}{b=0.1}&99.64&99.52&98.86&99.54&99.33&98.48&92.40&89.30&80.67\\\hline
\multicolumn{1}{r||}{b=0.4}&99.45&99.10&97.62&99.39&98.92&96.80&99.21&98.16&93.47\\\hline
\multicolumn{1}{r||}{b=0.9}&91.58&80.55&52.72&91.00&76.98&38.45&76.50&44.73& 9.57\\\hline
\end{tabu}
\)
\end{adjustbox}
\end{center}
\caption{Size corrected power (in $\%$) for independent $N(0,1)$-distributed innovations.}
\label{emppower}
\vspace{0.5cm}

\textbf{DOM kernel:}
\vspace{-0.3cm}
\begin{center}
\begin{adjustbox}{max width=0.9\textwidth}
\(
\begin{tabu}{l||c|c|c||c|c|c||c|c|c|}
\multicolumn{1}{c||}{\mbox{\textbf{Monitoring}}}&\multicolumn{3}{c||}{\boldsymbol{\beta=0.25,\quad k^{*}=3}}&\multicolumn{3}{c||}{\boldsymbol{\beta=1,\quad k^{*}=100}}&\multicolumn{3}{c|}{\boldsymbol{\beta=1.4,\quad k^{*}=630}}\\
\multicolumn{1}{c||}{\mbox{\textbf{scheme}}}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}\\\hline
\mbox{\textbf{CUSUM}}&98.48&97.96&96.66&97.87&97.08&94.78&88.33&84.07&74.53\\\hline
\mbox{\textbf{Page-CUSUM}}&98.39&97.73&96.17&98.01&97.12&94.85&92.82&88.49&78.21\\\hline
\mbox{\textbf{mMOSUM}}&&&&&&&&&\\
\multicolumn{1}{r||}{b=0.1}&98.26&97.64&95.94&98.22&97.43&95.33&91.62&87.66&78.42\\\hline
\multicolumn{1}{r||}{b=0.4}&97.17&95.48&89.94&97.12&95.30&89.02&96.43&93.80&84.27\\\hline
  \multicolumn{1}{r||}{b=0.9}&26.30&16.77&12.13&25.03&13.99& 9.03&14.42& 7.78& 7.45\\\hline
\end{tabu}
\)
\end{adjustbox}
\end{center}
$ $\\
\textbf{Wilcoxon kernel:}
\vspace{-0.3cm}
\begin{center}
\begin{adjustbox}{max width=0.9\textwidth}
\(
\begin{tabu}{l||c|c|c||c|c|c||c|c|c|}
\multicolumn{1}{c||}{\mbox{\textbf{Monitoring}}}&\multicolumn{3}{c||}{\boldsymbol{\beta=0.25,\quad k^{*}=3}}&\multicolumn{3}{c||}{\boldsymbol{\beta=1,\quad k^{*}=100}}&\multicolumn{3}{c|}{\boldsymbol{\beta=1.4,\quad k^{*}=630}}\\
\multicolumn{1}{c||}{\mbox{\textbf{scheme}}}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}&\boldsymbol{\gamma=0}&\boldsymbol{\gamma=0.25}&\boldsymbol{\gamma=0.45}\\\hline
\mbox{\textbf{CUSUM}}&100.00&100.00&100.00&100.00&100.00&100.00& 99.69& 99.37&97.93\\\hline
\mbox{\textbf{Page-CUSUM}}&100.00&100.00&100.00&100.00&100.00&100.00&100.00& 99.99&99.86\\\hline
\mbox{\textbf{mMOSUM}}&&&&&&&&&\\
\multicolumn{1}{r||}{b=0.1}&100.00&100.00&100.00&100.00&100.00&100.00& 99.93& 99.89&99.25\\\hline
\multicolumn{1}{r||}{b=0.4}&100.00&100.00&100.00&100.00&100.00& 99.99&100.00&100.00&99.98\\\hline
\multicolumn{1}{r||}{b=0.9}& 99.81& 98.82& 88.25& 99.77& 98.56& 83.37& 99.24& 92.86&50.94\\\hline
\end{tabu}
\)
\end{adjustbox}
\end{center}
\caption{Size corrected power (in $\%$) for independent $t_3$-distributed innovations.}
\label{emppower.t}
\end{table}

\begin{center}
\begin{figure}[tb]
\begin{tabular}{llll}
\hspace{-0.3cm}\vspace{-0.5cm}&\multicolumn{1}{c}{$\beta=0.25$}&\multicolumn{1}{c}{$\beta=0.75$}&\multicolumn{1}{c}{$\beta=1.4$}\\
\multicolumn{1}{l}{\raisebox{18ex}[0ex][0ex]{\small$\gamma=0$}}&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_1_1_v2.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_3_1_v2.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_6_1_v2.pdf}\\
\multicolumn{1}{l}{\raisebox{18ex}[0ex][0ex]{\small$\gamma=0.45$}}&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_1_3_v2.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_3_3_v2.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_6_3_v2.pdf}\\
\end{tabular}
	\caption{Estimated densities of the stopping time for the DOM kernel (solid lines) and the Wilcoxon kernel (dashed lines) for independent $N(0,1)$-distributed observations.}
	\label{run1}
\vspace{0.2cm}
\begin{tabular}{llll}
\hspace{-0.3cm}\vspace{-0.5cm}&\multicolumn{1}{c}{$\beta=0.25$}&\multicolumn{1}{c}{$\beta=0.75$}&\multicolumn{1}{c}{$\beta=1.4$}\\
\multicolumn{1}{l}{\raisebox{18ex}[0ex][0ex]{\small$\gamma=0$}}&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_1_1_tdist.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_3_1_tdist.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_6_1_tdist.pdf}\\
\multicolumn{1}{l}{\raisebox{18ex}[0ex][0ex]{\small$\gamma=0.45$}}&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_1_3_tdist.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_3_3_tdist.pdf}
&\hspace{-0.3cm}\includegraphics[width=0.3\textwidth]{runlength_6_3_tdist.pdf}\\
\end{tabular}
	\caption{Estimated densities of the stopping time for the DOM kernel (solid lines) and the Wilcoxon kernel (dashed lines) for independent $t_3$-distributed observations.}
	\label{run1.t}
	\end{figure}
	\end{center}

	Table \ref{emppower} reports the size-corrected power (i.e.\ the power that is achieved if the critical values are chosen such that all procedures have empirical size $\alpha$) while Figures~\ref{run1} and \ref{run1.t} show a density plot of the size-corrected run length that is scaled in such a way that the area under the curve integrates to the empirical power.

As has already been observed for the CUSUM statistic in \cite{kirch2018modified}   $\gamma=0$ has the highest power in all situations, where the superiority of $\gamma=0$ is particularly strong for late changes ($\beta=1.4$). In terms of detection delay the CUSUM statistic is almost always outperformed by the other monitoring schemes even for early changes.
 The modified MOSUM with $b=0.9$ has a very poor power when using $\gamma=0.45$, in particular for late changes. The peak at the beginning of the monitoring period in Figure \ref{run1} seems somewhat surprising as we already wait for $a_m=10$ observations before we start monitoring. However, it turns out that it relates to only $5\%$ false positives out of all simulation runs.
The modified MOSUM with $b=0.4$ has the most stable power with respect to the time of the change for both kernels. It also has generally the smallest detection delay with the exception of very late changes where it is outperformed by the modified MOSUM with $b=0.9$ and $\gamma=0$.

For early ($\beta=0.25$) and medium-late changes ($\beta=1$) the detection delay for $\gamma=0.45$ is somewhat smaller than for $\gamma=0$ despite the fact that the power is smaller in this case.
This indicates that early changes are detected either very quickly or not at all for $\gamma\neq 0$, whereas for $\gamma=0$ it might take a bit longer but the change will be detected at some point with a very high reliability.

Furthermore, the Wilcoxon kernel almost achieves the same power and detection delay as the DOM kernel for normal data, but clearly outperforms the latter in case of more heavy-tailed $t$-innovations (with $3$ degrees of freedom).
An additional benefit of the Wilcoxon kernel is its superiority with respect to robustness against outliers as can clearly be seen in Table 6.5 in \cite{diss} and in the density plots of the stopping times in Figure 6.3 in \cite{diss}.

\clearpage
\subsection{Data analysis}\label{sec.data}

\begin{figure}[tb]
Monthly \textbf{mean} temperature:
\begin{center}
\vspace{-0.5cm}
\includegraphics[width=0.8\textwidth]{tempmean_m120.pdf}
\end{center}
Monthly \textbf{minimal} temperature:
\vspace{-0.5cm}
\begin{center}
\includegraphics[width=0.8\textwidth]{tempmin_m120.pdf}
\end{center}
\caption{Deseasonalized monthly \textbf{mean} and \textbf{minimal} temperature in Potsdam, Germany. In both cases, the upper plot  shows the data, as well as the two change points detected by an a posteriori procedure, where the second one has been dominating. The lower plot shows the normalized monitoring statistics where monitoring is stopped as soon as the horizontal line at one is crossed. The vertical lines indicate when this happens. The Wilcoxon monitoring statistics are indicated by the dashed line, the DOM by a solid line. The blue line indicates the CUSUM monitoring, green Page CUSUM and red mMOSUM.}
	\label{mintemp}
	\end{figure}

In the following data example we analyze the monthly mean and minimal temperatures in Potsdam, Germany \footnote{provided by DWD at {https://opendata.dwd.de/climate\_environment/CDC/observations\_germany/climate/monthly/kl/historical/}} from 1893 to 2018. We use the first 120 months as our historic data set, thus starting the monitoring in 1913. In a preprocessing step we remove seasonalities by substracting the average temperature by month obtained from the historic data set. We then apply the different monitoring schemes with the DOM- as well as the Wilcoxon kernel to both data sets using the weight function in \eqref{exw} with $\gamma=0$. We test at a level of $\alpha=5\%$ and use $b=0.4$ for the modified MOSUM procedures.  For a better comparison the monitoring statistics are divided by the respective critical value and the weight function such that the null hypothesis is rejected as soon as the normalized monitoring statistic crosses the horizontal line at 1. The resulting rejection times as well as the normalized monitoring statistics can be seen in Figure~\ref{mintemp}. The black lines represent the change points resulting from the offline Wilcoxon procedure provided in \cite{Dehl}. First, the changes in October 1987 (mean temperatures) respectively in August 1987 (minimal temperatures) have been detected with p-values less than $5 \cdot 10^{-7}$. Splitting the data sets in those points and reapplying the procedure reveals additional changes in the first part of the data sets in April 1929/ August 1924 and in the second part in May/ June 2006 with p-values between $0.0004$ and $0.094$. For the monthly mean temperatures the mMOSUM in combination with the DOM kernel has a much smaller detection delay than all other procedures. Furthermore, the sequential procedures based on the DOM kernel detect the first change whereas the procedures based on the Wilcoxon kernel only reject after the second and more significant change has appeared. Opposite behavior, i.e. superiority of the Wilcoxon kernel, with an even stronger effect can be observed when analyzing the monthly minimal temperatures.
\vspace{2mm}

In conclusion, the present data example confirms the findings from the simulation study: On the one hand, the DOM kernel performs better for the mean monthly temperatures which are closer to normality. On the other hand, the Wilcoxon kernel is advantageous for the minimal monthly temperatures which are further away from normality and do contain more outliers. Additionally, for a given kernel the modified MOSUM has the smallest detection delay in all cases.

\clearpage

redpajama_set_name: RedPajamaArXiv


text:
Hamilton cops seize $3 million in drugs during trafficking probe
The 16 suspects are charged with a combined total of 213 offences
Brad Hunter
Some of the haul pulled in by Hamilton cops. Photo by SCREENGRAB /HAMILTON POLICE
Hamilton Police have arrested 16 people and seized millions in drugs, along with guns and cash, following a three-month probe.
In executing search warrants at six Hamilton Mountain properties, officers seized more than $3 million in drugs and cash, along with 35 firearms and thousands of rounds of ammunition as part of an investigation into drug trafficking and gun violence.
Dubbed Project Munro, the probe pulled a slew of firearms off the streets — 35% of the guns were either loaded or unsafely stored, according to police. A number of the guns were also modified to enhance performance.
In addition, three heavily plated high-calibre bulletproof vests were also seized.
In total, 24 search warrants were executed and 16 suspects charged, facing a combined total of 213 offences.
"Hamilton Police continue to relentlessly pursue offenders who are engaged in violence, drug activity and firearms in our community," cops said in a press release.
Over the past year, police in Ontario have seized tens of millions of dollars' worth of plants and products from large-scale illicit cannabis grow ops and production sites across the province.
The investigation remains ongoing, therefore further information was not released.
Meanwhile, on Thursday a man was taken to hospital and two others were arrested following a shooting near York Rd. and Hwy. 6. It's believed the injured man who suffered non-life threatening injuries was driving a tow truck.
The cops cordoned off the area. The canine unit and collision reconstruction unit were on scene.
Also, Hamilton Police arrested three males on gun and drug charges following a road incident last weekend.
On Saturday just after 3:30 a.m., cops on patrol in the area of Concession St. and Upper Gage spotted a vehicle being driven erratically.
One vehicle immediately fled when police were seen and it was located minutes later abandoned in the Upper Wellington St. and Limeridge R. E. area.
A loaded handgun and magazine were recovered in the vehicle, cops say. Moments later, the three suspects that had been in the vehicle were arrested a short distance away.
Shola Sadiq, 26, of Brampton, in charged with a number of firearm-related offences, possession over $5,000, fail to comply with an undertaking, dangerous operation of a vehicle, and a number of other charges.
Mohamed Fuad, 24, of Hamilton, is charged with a number of firearm-related offences, possession over $5,000, four counts of breach of a court order, and two counts of breach of probation.
A 17-year-old boy from Scarborough is charged with a number of firearm-related offences and possession over $5,000.
In other news,
bhunter@postmedia.com

redpajama_set_name: RedPajamaCommonCrawl


text:
New Survey Methods to Measure Quality of Life in Huntington's Disease Patients
Cutting-edge patient-centered "smart tests" will allow clinicians to evaluate specific quality of life concerns for individuals with Huntington's disease.
Author | Kylie Urban
Huntington's disease is an inherited disease that leads to problems with movement, thinking and mood which can negatively affect an individual's quality of life.
SEE ALSO: 'Game-Changing' Alzheimer's Drug Just a First Step
The fatal disease progressively causes nerve cell degeneration in the brain. Typically, individuals with Huntington's disease develop symptoms during their 30s or 40s, and these symptoms gradually worsen until death.
"The cognitive, behavioral and motor symptoms associated with Huntington's disease can significantly impair an individual's ability to perform their daily routine or lifestyle," says Noelle Carlozzi, Ph.D., associate professor of physical medicine and rehabilitation and director of the center for Clinical Outcomes Development and Application at the University of Michigan.
Carlozzi is the lead author of four new papers which present the results of her National Institutes of Health-funded study to develop new patient-reported outcome measures for Huntington's disease.
Patient-reported outcome measures allow clinicians to identify, understand and evaluate an individual's most important health concerns and whether current treatments are working. An example of a patient-reported outcome is when a patient is asked to describe his or her level of pain to a physician on a scale of 1 to 10, with 1 being little to no pain and 10 being the most pain the patient has ever felt.
Many patient-reported outcome measures exist, but measures are often geared toward the general population or other diseases and don't capture the specific symptoms, symptom burden and challenges that people face living with Huntington's disease.
"Although we have several ways to measure changes in health, for example through blood tests or magnetic resonance imaging [MRIs], these tests do not necessarily capture patient concerns," Carlozzi says. "My work is focused on developing patient-centered outcome measures that can capture the effect a disease has on an individual's physical, mental and/or social health."
'Smart tests' tailored to the patient
Carlozzi's four papers appear in the October 2016 print issue of Quality of Life Research journal and feature the Huntington disease quality of life (HDQLIFE) measurement system. All measures within this system were developed as computer-adaptive tests, or "smart tests," and are designed to relay information directly from patients to their clinicians about how Huntington's disease impacts their lives.
SEE ALSO: Improving Access to Epilepsy Care for Children in Michigan
Carlozzi and team held focus groups of patients with Huntington's disease, and separate groups for their family members and health care providers. The researchers asked several questions, such as how does having Huntington's disease affect quality of life, and identified the most important aspects of how the disease impacts social, emotional and physical well-being based on their responses.
"We used these discussions to identify the most important aspects of quality of life for these patients," Carlozzi says. "We looked at previously developed quality-of-life measurement scales that are not specific to Huntington's disease, called PROMIS [Patient-Reported Outcomes Measurement Information System] and Neuro-QoL [Quality of Life in Neurological Disorders], and identified aspects that were unique, as well as aspects from these measures that were relevant to individuals with Huntington's disease."
The team then used established measurement development procedures (which involved confirmatory and exploratory factor analysis and item response theory graded response models) to develop new patient-reported outcome measures for Concern with Death and Dying, Meaning and Purpose, Difficulties with Speech, Difficulties with Swallowing, and Chorea. They then took these five new measures, combined with general measures of quality of life relevant to individuals with Huntington's disease, to create the new HDQLIFE measurement system.
"This new measurement system will help clinicians monitor patients with Huntington's disease by detecting small, but important, changes in health-related quality of life," Carlozzi says. "Patient-reported outcome measures can allow for more efficient Huntington's disease clinical trials and enable patients to more effectively communicate their concerns to their providers during clinic visits."
All of these smart tests are individually tailored so the patient sees only questions or items that are the most relevant to him or her. The test administers the first item, and using calibration statistics, chooses the next best item based on the respondent's previous answer.
"Using this type of approach, respondents typically only answer four to eight questions per domain, and these measures only take one to two minutes to administer," Carlozzi says. "We're able to get an accurate response, without sacrificing sensitivity, and the patient doesn't have to sit and answer 50 to 100 questions."
She explains this is a positive move.
"We don't want to be asking an individual, especially one with cognitive problems, to take a long test. These findings will decrease the burden on patients and maximize efficiency for clinicians," she says.
Carlozzi hopes this work will improve patient and physician communication by giving patients a voice and a way to express their most troublesome symptoms.
"When you're working with a rare disease such as Huntington's disease, it is important that our measures are sensitive enough to detect subtle changes in functioning," she says. "Better measures mean we need fewer people in our clinical studies to know whether or not our treatments are working."
Participants were recruited at several locations in the USA to ensure a geographically diverse sample. This included eight established Huntington's disease clinics (UCLA; University of Iowa; Indiana University; Johns Hopkins University; University of Michigan; Struthers Parkinson's Center; Washington University; and Rutgers University) and the National Research Roster for Huntington Disease Patients and Families. Participants were also recruited in conjunction with other ongoing research studies, such as PREDICT-HD (UCSF; University of Iowa; University of Indiana; Johns Hopkins University; Washington University; and Cleveland Clinic), as well as in cooperation with Huntington's disease support groups and Huntington's disease specialized nursing home units (Phoenix, Arizona; Tuscon, Arizona; Denver, Colorado; Jacksonville, Florida; Des Moines, Iowa; Louisville, Kentucky; Lansing, Michigan; Robbinsdale, Minnesota; Lakewood, New Jersey; Plainfield, New Jersey; New York, New York; Dallas, Texas; and Seattle, Washington). Data Analysis was conducted at University of Michigan and Northwestern University.
Bryon Quertermous contributed to this report.
More Articles About: Industry DX Neurological (Brain) Conditions Physical Medicine & Rehabilitation
High fat diet activates early inflammation in mouse brains, supports link to neurologic disease
The findings may support the immune pathway as a bridge between diet and neurologic disease
4 things to know about stiff person syndrome
A neuromuscular specialist shares helpful facts about symptoms, diagnosis and treatment.


text:
World Map / Samoa Map
Samoa Map
More Maps of Samoa
Where is Samoa in the World
Samoa Outline Map
Samoa Latitude and Longitude Map
Samoa Cities Map
Samoa Road Map
Apia Map
La carte du Samoa
The Samoan islands are located in the South Pacific Ocean, and have been inhabited by humans for about 3,000 years. The early people of Samoa probably arrived from Southeast Asia as early as 2500 BCE, though there have been strong ties between other South Pacific islands, including Fiji and Tonga.
The Dutch were the first European explorers to arrive on the islands, in 1722, and were soon followed by the French in 1768. The English began sending missionaries in the early 1800s, and later sent troops. Germany and the United States also grew interested in the islands, and the latter three nations became involved in the local politics, sparking a civil war among the Samoan people from 1886 to 1894. A second civil war erupted in 1898, with the three foreign nations warring over their own claims to the land. The Tripartite Convention of 1899 resulted in the partitioning of the islands between the foreign powers. Under this pact, the eastern Samoan islands became US territories, eventually becoming part of American Samoa. The western islands became German Samoa, and the United Kingdom gave up their rights to the islands to Germany.
The New Zealand military arrived in German Samoa in 1914, taking control until 1962. The people of Samoa began their fight against colonial rule in the 1920s, however, and Western Samoa was granted independence in 1962. Samoa became the first independent island nation in the Pacific.
Neighboring Countries :
The closest islands to Samoa include the island territories of American Samoa, Tokelau, Wallis and Futuna, Niue, Fiji, and Tonga.
Geography :
The main islands of Samoa include Upolu and Savai'i, where most of the country's land is and the majority of the population resides, but there are also 8 small islands within the territory. Three of these, Manono, Apolima, and Nu'ulop, are located in the Apolima Strait, and four are the Aleipata Islands, which are Nu'utele, Nu'ulua, Namua, and Fanuatapu. The other islet of Samoa is Nu'usafe'e which is to the south of Upolu.
The islands of Samoa are volcanic islands, created by a hotspot volcano, and Savai'i island features active volcanoes, Mount Matavanu, Mata o le Afi, and Mauga Afi. Though these are considered active, there have been no eruptions since the early 1900s. These eruptions have left lava fields across the island. As volcanic islands, the terrain is mountainous, with its highest point on Mount Silisili, standing 1,858 meters (6,096 feet) above sea level.
The islands were once covered with tropical forests, but with human activity on the island, the original forests have largely been destroyed. The islands continue to be covered in younger forests and other vegetation. Samoa has no major rivers, though there are many streams, some of which are prone to flooding in times of heavy rainfall. A major stream is the Vaisigano river.
Points of Interest :
Samoa is full of scenic attractions, including national parks like Falealupo Rainforest Preserve and Lake Lanoto'o National Park. There are wonderful hiking opportunities in the mountains of the main islands, including impressive waterfalls like those at Papase'ea Sliding Rocks on Upolu. As a volcanic island chain, there are several natural features that have been formed, including lava fields and caves and blowholes where the ocean waves shoot up between the rocks.
The beaches of Samoa are a beloved destination for visitors and locals alike, and can be accessed from most villages on the islands. There are also many resorts along the coast.
The main airports in Samoa are Faleolo International, which is outside of Apia, and Fagalii Airport, also outside of Apia. Main destinations served by these airports include New Zealand, Australia, Fiji, and the United States. Boat is an alternative way to reach Samoa, with major harbors in Apia, Asau, Mulifanua, and Saleloga. Boats are also the main way to get between the islands. Taxis are a great option for getting around the islands as they are everywhere and can be relatively inexpensive. Another option is the bus, which provide cheap and efficient service. Visitors can rent cars or potentially bicycles to get around the islands as well.
Area(km.&sup²)
Area(mi.&sup²)
Eastern 23,639 67 26 Aunuu, Tutuila (part)
Manu'a 1,143 57 22 Ofu, Olosega, Tau
Western 30,720 75 29 Tutuila (part)
Unorganized 17 2 1 Swain's Island, Rose Island
4 divisions 55,519 200 77
Note: Population: 2010-04-01 census.
Pop-2006
Area(km.²)
Area(mi.²)
A'ana 20,769 18,000 193 75 Leulumoega Upolu
Aiga-i-le-Tai 4,857 4,000 27 10 Mulifanu'a Upolu
Atua 21,826 18,000 413 159 Lufilufi Upolu
Fa'asaleleaga 13,404 10,000 266 103 Safotulafai Savai'i
Gaga'emauga 7,487 6,000 223 86 Sale'aula Savai'i, Upolu
Gagaifomauga 4,842 5,000 365 141 Safotulafai Savai'i
Palauli 9,082 8,000 523 202 Vailoa i Palauli Savai'i
Satupa'itea 5,260 4,000 127 49 Satupa'itea Savai'i
Tuamasaga 85,112 51,000 479 185 Afega Upolu
Va'a-o-Fonoti 1,624 2,000 38 15 Samamea Upolu
Vaisigano 6,478 6,000 178 69 Asau Savai'i
11 districts 180,741 132,000 2,832 1,094
Note : -Pop-2006: 2006-11-06 census.
ACOD~20131023
Last Updated On : July 29, 2015
About Samoa
Where is Apia
Apia, Capital of Samoa
Samoa Geography & History
Culture in Samoa
Samoa Clothing
Education in Samoa
Weather in Samoa
Population in Samoa
Embassies and Consulates of Samoa
Cultural Festivals
Samoa Travel
Tourist Attractions in Samoa
Hotels in Samoa
Map of Australia
Fiji Map
Micronesia Map
Papua New Guinea Map
Solomon Islands Map
Vanuatu Map
Guam Map
Kiribati Map
Marshall Islands Map
Nauru Map
Northern Marianas Map
Palau Map
French Polynesia Map
Cook Islands Map
Niue Map
Pitcairn Islands Map
Tonga Map
Tuvalu Map
Norfolk Island Map
Samoa Flag
The Samoa flag was officially adopted on January 1, 1962 when it gained independence.However, Samoa had adopted a same flag like the present one on 26 May, 1948.
Facts about Samoa
ContinentOceania
Lat Long13.7745° S, 172.0428° W
Area2,831 sq km (1,093 sq miles)
Population195,476 (July 2013 est.)
CapitalApia
Largest CityApia
Official LanguagesSamoan, English
Major ReligionProtestant, Roman Catholic etc.
National DayIndependence Day (1 June, 1962)
Form of GovernmentParliamentary Democracy
PresidentO le Ao o le Malo Tufuga Efi
Prime MinisterTuilaepa Aiono Sailele Malielegaoi
CurrencyTala (WST)
GDP$6,200 (2012 est.)
Calling Code685
Time ZoneUTC+13:00
Internet TLD.ws


text:
To solve the pressing problems of our field, we pursue shared research with a host of government and industry partners.
The Collaboration Destination
The Department of Chemical and Biomolecular Engineering lives on NC State's Centennial Campus, an award-winning research environment that is home to more than 70 government and industry partners. Our neighbors include some of the best in business; our faculty seek out mutually beneficial relationships with companies big and small.
Those relationships include partnering on Small Business Innovation Research or Short Term Innovation Research grants; working with our centers, institutes and laboratories; providing materials and facilities to the department; or directly funding our students, postdoctoral fellows and research projects.
Interested partners are encouraged to contact the department or get in touch with our faculty directly.
Bright Minds, Brought Together
Faculty in the Department of Chemical and Biomolecular Engineering hold leadership positions in some of NC State's many leading research centers, institutes and laboratories. Opportunities abound to collaborate with us at some of the top research and training facilities in the nation.
Ruben Carbonell is the director of the Kenan Institute and the Biomanufacturing, Training and Education Center (BTEC).
Many faculty in the department are also key participants in BTEC, including Stefano Menegatti and Henry Lamb.
Robert Kelly is the director of NC State's Biotechnology Program.
Wesley Henderson (through the Ionic Liquids and Electrolytes for Energy Technologies Laboratory) and Peter Fedkiw work with the Advanced Transportation Energy Center to develop batteries for plug-in hybrid electric vehicles.
Keith Gubbins and Carol Hall utilize the resources of the High-Performance Computing Center and Keith Gubbins is a co-director of the Center for High Performance Simulation.
Keith Gubbins also helped to found the Institute for Computational Science and Engineering, of which Phillip Westmoreland is the director.
Greg Parsons has been instrumental in organizing NANO@NC STATE, which brings together many of the department's and university's nanotechnology researchers.
Behnam Pourdeyhimi is the executive director of the Nonwovens Institute.


In addition to providing startup and established managers with information on various hedge fund laws and regulations, we have also provided a number of articles on the general business issues that hedge fund managers need to address. These articles cover a number of areas and many of the articles below are from service providers to hedge funds. Please feel free to suggest other areas for us to explore in future articles.
IT Issues for Hedge Fund Managers – this article, written by a technology solutions firm, details some of the IT solutions which are available to hedge fund managers who will need to register after the new hedge fund registration rules are adopted. These issues also apply for currently registered (SEC or state) hedge fund managers as well.
Data Warehouse Implementation for Hedge Funds – a summary of a panel discussion on the reasons managers should think about implementing a data warehouse system.
Hedge Fund Hotels – hedge fund hotels offer managers a whole suite of hosted services for turnkey back office implementation.
Hedge Fund Domain Names – this article discusses domain names for hedge fund managers.
Hedge Fund Managers and Blogs – this article discusses whether hedge fund managers can run a blog.
Proposition Q and Hedge Funds – hedge fund managers located in San Francisco should be aware of Proposition Q (a special San Francisco payroll tax) and how it will affect them and their top earners.
Hedge Fund Employment Law Issues – this article, forthcoming, will discuss the issues that managers need to consider when they bring on employees for the first time. While each state will have different laws and regulations which the manager will need to follow, there are general items which a manager should be prepared for including establishing a structure to pay state and federal income taxes for employees.
IP Licensing Agreements – this article, forthcoming, will discuss the business and legal issues involved when managers license their intellectual property to a hedge fund.
Hedge Fund Management Company Insurance – this article discusses the costs of insurance for hedge fund management companies.
Hedge Funds and Rehypothication – this article discusses rehypothication as it relates to hedge funds.
Prime Brokers, Margin Lock-ups and Hedge Funds – this article discusses prime broker margin lock-ups and some of the issues which managers should understand.
Hedge Fund Operational Issues and Failures – this article discusses a press release about a white paper released from a hedge fund due diligence firm. The firm examines the reason why many hedge fund managers fail.
Hedge Fund Operations During a Pandemic – examines the issues which face managers because of the H1N1 virus.
Agreed Upon Procedures – hedge fund managers sometimes will engage auditors to provide "agreed upon procedures" especially with regard to their Level III assets. This article, written by a hedge fund due diligence firm, provides some thoughts on how investor may view these procedures in the future. Managers should be aware of how investors (or potential investors) might view these types of engagements.
Women and Hedge Funds – this article explores the world and opportunities for women in the hedge fund industry.
Naming Your Hedge Fund – this article discusses common hedge fund naming conventions.
Hedge Fund Best Practices – discussion of the President's Working Group on Financial Markets (PWG) report on hedge fund best practices.
SIPC Customer Asset Protection – statements by FINRA and the SEC on asset protection for customers.
This entry was posted in Business Issues and tagged hedge fund, hedge fund business issues, hedge fund insurance, hedge fund IT, hedge fund IT solution, hedge fund technology on October 29, 2009 by Hedge Fund Lawyer.


A word has recently been coined in the women's health arena and we're hearing it more and more these days. That word is "preconception." Preconception care has become a rather hot topic and one that has been a long time coming into its own. The idea of preconception care carries with it the import of what needs to be done before conception and, as such, it carries a great deal of weight.
There are many considerations a woman must make if she is planning her pregnancy. In truth, pregnancy often happens without planning and many mothers-to-be weren't sure they wanted what they were getting. When it comes to juggling life, careers, finances and relationships, these days preconception thinking is a prudent measure. Preconception planning is more than figuring out when you want to conceive and, for women of reproductive age, it is now recognized as a critical component of health care.
The idea behind preconception care is to make available the important screening, testing, health initiatives and interventions for women who want to become pregnant before they do so, in order to reduce the risks that may endanger a future pregnancy. This system seeks to identify potential risk factors, whether they are biomedical, behavioral or social, that may influence a future pregnancy and to provide prevention and management strategies. These strategies require action before conception or very early into the pregnancy in order to be effective.
On one level, many women are very aware of the need to be on top of their health before they become pregnant. However, it remains a stunning fact that in the United States, despite the major advances in medical care, poor birth outcomes continue to be staggering. Consider some of these statistics: Each year, 12 percent of babies are born premature, 8 percent are born with low birth weight, and 3 percent have major birth defects. 31 percent of women giving birth encounter pregnancy complications. Even though there is abundant information available to the public, 11 percent of women continue to smoke during pregnancy and 10 percent continue to drink alcohol. 31 percent of women who conceive are obese and 3 percent take over-the-counter drugs and medications which could endanger the life of their unborn baby. 69 percent of women do not take folic acid supplements which have been shown to be a crucial factor in the health of the forming baby. Finishing the list is the 4 percent of women who go into a pregnancy with an unmanaged, pre-existing condition such as diabetes, which can be dangerous if it is out of control.
Prenatal care is too little too late when it comes to intervention in potentially problematic pregnancies. The fetus is susceptible to developing certain difficulties in the first 4 to 10 weeks after conception, a time when most women either don't know they're pregnant or have not seen a medical professional for prenatal care.
Preconception care is important to the future health of the nation, yet it is still a service that is not often offered to the majority of women of child-bearing age. When preconception becomes a national issue, then hopefully more women will have the advantage of taking proper steps before they conceive to ensure the health and safety of themselves and their yet-to-be-conceived babies.


text:
Devin Townsend - conquering a wilderness of rhythms and themes
Report by: Andreja Krušič
Photographs by: Milan Krušič
Location: Tvornica kulture, Zagreb, Croatia
A prolific musician that he is, Devin Townsend isn't the one to waste the opportunity to impress the capacity crown tonight in Zagreb, Croatia. Townsend does exactly that with a finely tuned set that goes down a treat on this cold Sunday night.
The six-piece bound on stage to the adrenalized anthems "Supercrush!", Night" and "Hyperdrive". Needless to say, Townsend is a dynamic focal point, his presence oozing total command of the stage, and his voice, a combination of shrieks and growls, sending shivers down the spine. Meanwhile, Dave Young prowls the stage, peeling off paintwork with monolithic guitar riffs.
The band's engine room, occupied by Ryan Van Poederooyen on drums and Brian Waddell on bass, is tight, propelling the band through a wilderness of rhythms and themes without a hic-up. Clearly the secret weapon of the band is its virtuosity, and this lot does well to exploit their assets.
There are numbers from "Transcendence"; "Stormbending" and "Failure" lining up to win over the audience. Fittingly, the evening ends with "Turth", another newbie, ensuring that nobody leaved the venue unsatisfied. Indeed, this is a night to remember.


text:
Kate Taylor (born August 15, 1949) is an American singer-songwriter, originally from Boston, Massachusetts.  She is the younger (and only) sister of renowned singer-songwriter James Taylor, a six-time Grammy winner.

Biography
Taylor was born in Boston and grew up with her four brothers in Chapel Hill, North Carolina, where her father Isaac Taylor was dean of the medical school at the University of North Carolina. Her mother, Trudy, grew up in Newburyport, Massachusetts, and received training as a lyric soprano in Boston.

Taylor formed her first band at age 15 and had her first record deal with Atlantic Records four years later. Produced by her manager at the time, Peter Asher, she released her debut album Sister Kate in 1971. Her second album, the 1978 self-titled Kate Taylor, was produced by her brother James and Lew Hahn on Columbia Records.

The following year Taylor released It's in There, produced by Barry Beckett at the famed Muscle Shoals studios.

Discography
Sister Kate (1971)
Kate Taylor (1978)
It's in There (1979)
Beautiful Road (2003)
Kate Taylor Live at The Cutting Room (2005)
Fair Time!  (2009)
Why Wait! (2021)

Participations on compilation albums
Strong Hand of Love, tribute to Mark Heard, 1994
Orphans of God, tribute to Mark Heard, 1994
Follow That Road, 1994
In Harmony: A Sesame Street Record, 1980

References

External links
Kate Taylor official website

1949 births
Living people
American women singers
Songwriters from Massachusetts
Musicians from Boston
James Taylor
Taylor family (show business)

redpajama_set_name: RedPajamaWikipedia


Shipping $14.34 Add item to cart for shipping estimate. You can SAVE on shipping by buying multiple items from the same seller. Shipping is charged by each individual seller. Sellers at times may also provide FREE SHIPPING on select items or on entire order if it meets the minimum order value. Please check sellers page for more details.
Extra Large size Iron Tava / Pan with Handle. Ideal for or making Roti, Chapati, Iron, paw bhaji etc at restaurants or at catering events. This heavy gauge Iron tawa is anti-rust coated which gives it a clean look. The attached handle makes it easy to move the tava when hot.
Good for making multiple small rotis at the same time or paw bhaji in your restaurant or food stall.
Available Size :- 14 inch, 16 inch, 18 inch, 20 inch and 24 inch diameter.
Tava is curved with .5 to 1" deep in the center.
Shipping Cost $14.34 Buy multiple items online from Sangu Enterprise Ltd to SAVE on shipping.
This item in Cookware & Appliance category was first added to our catalog on 08-03-2015.
Buy Large Tawa w/ Handle, Iron Roti / Pav Bhaji Tava (14" - 24"), sold & shipped by Sangu Enterprise Ltd where you can shop online for Indian Restaurant Supply, Pooja Mandir, Handicrafts, Utensils & Gifts.
This Cookware Roti Tawa collection has been tagged as Roti Coooking Pan, Iron Cookware, Tawa, Dosa Roti Paratha Chapati Paw Bhaji Maker, Extra Large Tawa Pan, Indian Restaurant Supply, Tawa For Restaurant.
Can I use this for dosa?
This tawa is curved. For dosa, you need a flat tava.
What is the shipping charge to Niagara Falls, NY?
Thanks for your inquiry. Once you add the item to the shopping cart, it will show you the shipping charge for any destination in the USA.
This tawa is not meant for making dosas as it is curved. You will need a flat tava for dosas.
The largest tawa we have in stock is 24" diameter.
This Tawa using also dosa too ?
What is the availability for the 24 inch Tawa?
24 Inch Iron Tawa With Handle is available in stock.
Stainless Steel Curved Large Tava / Grill - 18" - 30"


text:
Anthony Turpel talks starring in "Love, Victor" – OUT TODAY!!!
Mordechai Laub
BlogFeaturedLove, VictorTV Shows
The highly awaited series "Love, Victor" is premiering TODAY on Hulu! The show is based off of the hit movie (one of our favorites) "Love, Simon." (Which was based off of the best-selling novel "Simon Vs. The Homo Sapiens Agenda.") "Love, Victor" follows the life of Victor who moved to a new neighborhood, is attending a new school and is realizing his sexuality. His journey as part of the LGBT+ community is sure to inspire viewers. In his new town, Victor befriends his neighbor Felix, played by Anthony Turpel. And Anthony gave a peek at what fan's can expect from the new show!
Young Entertainment Mag: You're starring in the new Hulu show "Love, Victor," which is loosely based on the book (Simon vs. The Homo Sapiens Agenda) and movie "Love, Simon." Were you familiar with "Love, Simon" before joining the show?
Anthony Turpel: Yes, I actually went to the theater to see it and thought it was amazing. I thought the message was wonderful, it was well-written and well-acted. So when I got the audition to the show I was extremely excited because I loved the world it created and it pushed forward a really important message.
YEM: "Love, Victor" exists in the same universe as "Love, Simon." Will fans see characters or references to "Love, Simon" in this new show?
Anthony: Yes they do! I don't want to spoil too much but I'll say that you will see a lot of familiar faces.
YEM: "Love, Victor" tells the story of a member of the LGBT+ community as he navigates high school and grows into his own. How does it feel to be a part of such a groundbreaking show?
Anthony: Oh it feels incredible. I think more representation and stories like this need to be shown. And I'm very thankful that I get to be a part of it. I think this story will help a lot of young kids who are confused and it will be a great place to turn and not feel so alone.
YEM: How would you best describe Felix and how does he relate to someone in high school now?
Anthony: Felix is just a lovable guy who accepts people for who they are. He represents the kid who's not afraid to be himself.
YEM: What is the dynamic between Felix and Victor? What kind of friend is Felix?
Anthony: I see the dynamic as Felix had no friends and Victor had no friends and for that reason alone Felix took that as an opportunity to establish a relationship between the two. He's always there for you. He tries to come from a place of understanding and love and that's exactly what Victor needed.
(Photo by: Eric McCandless/Hulu)
YEM: Were you able to use anything from your personal high school experience?
Anthony: Funny you should say that: I was actually homeschooled. So besides being an awkward mess, no. You could say I'm living my high school experience right now through the show.
YEM: What has the response been so far? Have you heard from fans that are excited for the show to premiere?
Anthony: Yeah, it seems like people are really excited about the show. The response has been so strong because stories like these have been lacking. "Love, Simon" was a powerful coming-of-age story that didn't have a tragic ending. Especially in a time like this I think the show will resonate with a lot of people.
YEM: What has been one challenge acting in "Love, Victor" that you were able to overcome?
Anthony: I couldn't say the word "simpatico". It was so many takes that they didn't even include it in the episode.
YEM: Did you have any firsts on this project? Something new in your acting that you're able to bring to your next gig?
Anthony: It was the first time I shot a scene at a giant carnival. I also got to work with Andy Richter and he gave me some pretty unique dating advice. You know, every time you're on set it's always a learning experience. So no matter what I'm on I'm always learning more and more things. And this show was no exception.
"More representation and stories like this need to be shown"
YEM: Is there room for a Season 2?
Anthony: Hell yeah! I think there's definitely room for a season 2, 3, and 4.
"Love, Victor" is now streaming on Hulu!
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actor anthony turpel hulu Interview lgbt love simon love victor


We are currently only accepting new members that plan to actively participate in group events. Since we are a true hiking club, you must have a clear profile photo of yourself for the safety and quality of this club and for search and rescue reasons. We also don't accept screen names, fictitious names, initials, etc. We join these Meetup groups to do activities with likeminded individuals and to meet others. Using at least your "real" first name allows this club to have a personable and friendly atmosphere.
Welcome to the Idaho Hiking Club, led by experienced hikers who share your interest in hiking and the outdoors. Come join us on a fun-filled event where you will meet new people and explore new places.
Idaho Hiking Club is open to all ages and ability levels from beginners to advanced hikers. Children and anyone under the age of 18 years of age are welcome to participate at events (unless otherwise stated), as long as they are accompanied by a parent or legal guardian.
We hike locally, all over the state, and beyond. Though hiking is our main focus, we also offer camping, backpacking, snowshoeing and other outdoor activities… of course we like to throw in a few social events too. Come explore, discover and conquer the trails with us!


Yesterday, the Alibaba announced the 1919 strategic investment of 2 billion yuan. This means that Tmall or will be in the wine industry launched a new retail layout.
Alibaba said that the Alibaba investment after 1919 and Tmall, rookie, ant gold clothing, Ali Ali cloud digital economy sectors, will carry out in-depth cooperation in the new retail, supply chain, warehousing and distribution, finance, data resources and so on. Tmall big FMCG division general manager Hu Weixiong said, the Alibaba is promoting the new retail strategy, to promote the upgrading of business and digital data. Have a drink 1919 new retail model, completed the integration of online and offline business system, the cooperation between the two sides will reconstruct the line wine and drink consumer scene, provide a better experience for the user to purchase wine.
Drink wine industry online penetration rate of less than 5%, 1919 in the field of online and offline retail liquor have depth layout, current online business electricity supplier industry ranked the top three, 1919 drink APP annual sales of 2 billion yuan, with 19 minutes to speed up the delivery service. At the same time, 1919 in the country with 1200 straight Direct stores, is currently promoting the creation of warehouse stores next door.


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Volume 69, Issue Suppl 2
P189 Should Screening For Chronic Viral Hepatitis In Patients With Tuberculosis Be Introduced To Nice Guidelines?
TB: non pulmonary and hepatotoxicity
JL Potter,
C Hyams,
M Shaukat,
ZO Babiker,
VM Macavei,
N Jayasekera,
H Kunst,
GR Foster,
VLC White
Barts Health NHS Trust, London, UK
Background Screening for viral hepatitis is not routinely recommended in patients diagnosed with tuberculosis (TB). However there are significant similarities in the global distribution of TB and hepatitis B (HBV) and C (HCV). It remains unclear whether co-infection with HBV or HCV is a risk factor for hepatotoxicity in patients receiving anti-tuberculous therapy and significant morbidity and mortality is associated with a late diagnosis.
Objectives To determine the prevalence of HBV and HCV infection among new cases of active TB across treatment centres in East London and to assess the adverse drug reactions to anti-tuberculous treatment experienced by this population.
Methods We conducted a retrospective study including all patients diagnosed with active TB during 2013 at two TB clinics in London. Data on demographic characteristics, HBV surface antigen (HBsAg), HCV antibody, human immunodeficiency virus (HIV) and adverse drug reactions were retrospectively analysed.
Results In total, 472 cases of active TB were notified during 2013. The mean age was 37.7 (+/- 15.3) years (range: 5–92). Males accounted for 62.3% of our cohort. 84.7% of patients were born outside of the UK with the majority of patients being born in either Bangladesh (16.5%), India (27.8%) or Pakistan (15.9%). Overall, 304 patients were screened for HBV, 302 for HCV, and 447 for HIV. Of those screened, HBsAg was detected in 3.3%, HCV antibody in 2.0% and HIV in 3.4%. All patients infected with HBV or HCV were foreign born. Hepatotoxicity was defined as an ALT greater than 5 times the upper limit of normal or requiring a change in treatment. There was no significant difference in rates of hepatotoxicity in either in HepBsAg status (p = 0.371), HCV status (p = 0.597) or HIV status (p = 0.413) but numbers of HBV and HCV infection were small.
Conclusions The prevalence of HBV and HCV was significantly higher in our cohort of TB patients than the background UK prevalence, which is 0.4% for HCV, 0.3% for HBV and 0.15% for HIV. Routine screening for HBV and HCV on an opt-out basis would be justified in our setting given the high proportion of foreign-born patients. Further research into the magnitude of HBV/HCV co-infection with active or latent TB, any increased risk in drug-induced hepatotoxicity and the cost-effectiveness of routine screening is needed.
http://dx.doi.org/10.1136/thoraxjnl-2014-206260.318
© 2014, Published by the BMJ Publishing Group Limited. For permission to use (where not already granted under a licence) please go to http://group.bmj.com/group/rights-licensing/permissions
Copyright © 2020 BMJ Publishing Group Ltd & British Thoracic Society. All rights reserved.


A tissue biopsy is often required to make a definitive diagnosis of lung cancer. However, because of small size and inadequate biopsy yield, early-stage lung cancer is often difficult to diagnose. Dr. Hariri is using a novel imaging technique called optical coherence tomography (OCT) to develop tools to guide tissue biopsy sampling to improve tissue yield. These tools will also provide additional diagnostic information.
Early, accurate diagnosis of lung cancer is fundamental to improving patient survival. Tissue biopsy is key to early diagnosis in order to determine if a lung nodule is malignant or benign. Unfortunately, low-risk methods of biopsy, such as bronchoscopy, are often not able to adequately sample targeted nodules when they are small and/or difficult to navigate to. If a diagnosis cannot be made, patients must undergo repeat biopsy or even surgery, which increases risk and delays diagnosis and therapy. This project aims to dramatically improve lung cancer diagnosis on low-risk biopsy using cutting-edge optical imaging tools in combination with navigation techniques to provide (1) real-time, intra-procedural assessment of biopsy site locations to ensure adequate tissue sampling and (2) large-volume "virtual optical biopsy" of nodules for diagnosis as a complement to tissue biopsy. This project will result in a powerful new bronchoscopy tool that could reduce unnecessary risky procedures, eliminate delays in diagnosis, and allow earlier therapy initiation.
Inadequate biopsy yield is a significant limitation to early lung cancer diagnosis and is a two-pronged problem: (1) Early-stage lung cancers are not adequately targeted during biopsy due to their small size and (2) even if a nodule is adequately targeted, often there is insufficient tumor in the biopsy to make a definitive diagnosis and/or accomplish all diagnostic testing for patient care. Optical coherence tomography (OCT) is a high-resolution imaging modality that provides virtual visualization of tissue volumes orders of magnitude larger than biopsy without tissue removal. Dr. Hariri's group has developed OCT catheters compatible with standard bronchoscopes and demonstrated that OCT can identify lung nodules and assess pathologic features of lung cancer with high sensitivity/specificity. This project's aims are to dramatically improve diagnostic capability in lung cancer using large volume OCT to provide (1) virtual intra-procedural evaluation in real-time (VIPER) of biopsy site location and 2) large volume virtual optical biopsy for subsequent pathology diagnosis as a complement to tissue biopsy. In Aim 1, they will assess if in vivo OCT VIPER biopsy assessment increases tumor yield on bronchoscopic biopsy in a powered, blinded clinical study. In Aim 2, they will use the data obtained in Aim 1 to determine the diagnostic accuracy of large volume OCT optical biopsy with tissue biopsy as compared to tissue biopsy alone. This project will result in clinical translation of a powerful, robust new optical bronchoscopy tool that could reduce unnecessary diagnostic procedures, eliminate delayed diagnoses, provide earlier intervention, and improve patient morbidity/mortality.


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Administering COVID-19 Vaccine to Susceptible Animals
Post on August 22, 2022
Tulsa Zoo has begun administering COVID-19 vaccines to zoo mammals that are considered susceptible to coronavirus. The vaccine, which is manufactured by Zoetis, is made exclusively for animals. The zoo is prioritizing species based on their risk for developing COVID-19, including tigers, lions, snow leopards, jaguars, bears, chimpanzees and other species.
"Throughout the pandemic, we established many precautionary measures to help ensure our animals remained safe," said Director of Animal Health and Senior Staff Veterinarian Dr. Kay Backues. "We're offering the vaccine as an additional form of preventative care, further protecting the animals in our care, including many species which are endangered."
Each animal will receive two doses, given three weeks apart. It is anticipated all identified animals will be vaccinated by the end of September. Tulsa Zoo animal care and veterinary staff are monitoring the animals closely for any signs of adverse reactions.
To create this vaccine, Zoetis' research and development team applied decades of experience developing other coronavirus vaccines for cats, dogs, poultry and cattle. Although the virus – or antigen – is the same as in human vaccines, this vaccine was developed only for animals.
"While additional studies are needed as we learn about this virus in zoo animals, we do know it can spread from people to animals in some situations, primarily from close contact with people who are infected with COVID-19. Based on the available information to date, the risk of animals spreading COVID-19 to people is considered very low," Dr. Backues said. "Tulsa Zoo implemented special protocols to protect susceptible animals, as we do for any zoonotic disease, meaning diseases that can be transmitted between humans and the animals in our care."
For example, the zoo limits behind-the-scenes access to certain animal areas and requires animal care staff working indoors in proximity to at-risk animals to wear respiratory protection.


Sofia and her family prepare for the Royal Prep Academy Graduation day!
Sofia continues to find out what being royal is all about! After discovering the fabled Mystic Isles, Sofia is invited to join a powerful group of adventurers who help protect the realm's magic and creatures. No matter where the adventure takes her, Sofia will prove she can conquer any challenge in her own way!"