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models constructed with @xmath0-stable distributions possess several useful properties , including infinite variance , skewness and heavy tails (; ; ; ) .
@xmath0-stable distributions provide no general analytic expressions for the density , median , mode or entropy , but are uniquely specified by their characteristic function , which has several parameterizations . considered as generalizations of the gaussian distribution ,
they are defined as the class of location - scale distributions which are closed under convolutions .
@xmath0-stable distributions have found application in many areas of statistics , finance and signal processing engineering as models for impulsive , heavy tailed noise processes (; ; ; ; ; ) .
the univariate @xmath0-stable distribution is typically specified by four parameters : @xmath1 $ ] determining the rate of tail decay ; @xmath2 $ ] determining the degree and sign of asymmetry ( skewness ) ; @xmath3 the scale ( under some parameterizations ) ; and @xmath4 the location . the parameter @xmath0 is termed the characteristic exponent , with small and large @xmath0 implying heavy and light tails respectively .
gaussian ( @xmath5 ) and cauchy ( @xmath6 ) distributions provide the only analytically tractable sub - members of this family . in general , as @xmath0-stable models admit no closed form expression for the density which can be evaluated pointwise ( excepting gaussian and cauchy members ) , inference typically proceeds via the characteristic function .
this paper is concerned with constructing both univariate and multivariate bayesian models in which the likelihood model is from the class of @xmath0-stable distributions .
this is known to be a difficult problem .
existing methods for bayesian @xmath0-stable models are limited to the univariate setting (; ; ; ; ; ) .
inferential procedures for @xmath0-stable models may be classified as auxiliary variable methods , inversion plus series expansion approaches and density estimation methods .
the auxiliary variable gibbs sampler increases the dimension of the parameter space from @xmath7 ( @xmath8 and @xmath9 ) to @xmath10 , where @xmath11 is the number of observations . as strong correlations between parameters and large sample sizes
are common in the @xmath0-stable setting , this results in a slowly mixing markov chain since gibbs moves are limited to moving parallel to the axes ( e.g. ) .
other markov chain monte carlo ( mcmc ) samplers ( ) adopt inversion techniques for numerical integration of the characteristic function , employing inverse fourier transforms combined with a series expansion to accurately estimate distributional tails .
this is performed at each iteration of the markov chain to evaluate the likelihood , and is accordingly computationally intensive .
in addition the quality of the resulting approximation is sensitive to the spacing of the fast fourier transform grid and the point at which the series expansion begins ( ) .
univariate density estimation methods include integral representations , parametric mixtures (; ) and numerical estimation through splines and series expansions (; ) .
approximates symmetric stable distributions using a mixture of gaussian and cauchy densities . and approximate the @xmath0-stable density through spline polynomials , and via a mixture of gaussian distributions .
parameter estimation has been performed by an expectation - maximization ( em ) algorithm and by method of ( partial ) moments ( ) . implemented within an mcmc sampler , such density estimation methods would be highly computational .
none of the above methods easily generalize to the multivariate setting .
it is currently only practical to numerically evaluate two dimensional @xmath0-stable densities via inversion of the characteristic function . here
the required computation is a function of @xmath0 and the number and spread of masses in the discrete spectral representation (; ) . beyond two dimensions this procedure becomes untenably slow with limited accuracy .    in this article
we develop practical bayesian inferential methods to fit univariate and multivariate @xmath0-stable models . to the best of our knowledge
, no practical bayesian methods have been developed for the multivariate model as the required computational complexity increases dramatically with model dimension .
the same is true of classical methods beyond two dimensions .
our approach is based on recent developments in `` likelihood - free '' inference , which permits approximate posterior simulation for bayesian models without the need to explicitly evaluate the likelihood .    in section [ sec : abc ] we briefly introduce likelihood - free inference and the sampling framework used in this article .
section [ sec : alpha - stable ] presents the bayesian @xmath0-stable model , with a particular focus on summary statistic specification , a critical component of likelihood - free inference .
we provide an evaluation of the performance of the proposed methodology in section [ sec : examples ] , based on controlled simulation studies in 1 , 2 and 3 dimensions .
finally , in section [ sec : realdata ] we demonstrate an analysis of real daily currency data under both univariate and multivariate settings , and provide comparisons with existing methods .
we conclude with a discussion .
computational procedures to simulate from posterior distributions , @xmath12 , of parameters @xmath13 given observed data @xmath14 , are well established ( e.g. ) .
however when pointwise evaluation of the likelihood function @xmath15 is computationally prohibitive or intractable , alternative procedures are required .
likelihood - free methods ( also known as _ approximate bayesian computation _ ) permit simulation from an approximate posterior model while circumventing explicit evaluation of the likelihood function (; ; ; ; ) .
assuming data simulation @xmath16 under the model given @xmath17 is easily obtainable , likelihood - free methods embed the posterior @xmath15 within an augmented model @xmath18 where @xmath16 , @xmath19 , is an auxiliary parameter on the same space as the observed data @xmath20 .
the function @xmath21 is typically a standard smoothing kernel ( e.g. ) with scale parameter @xmath22 , which weights the intractable posterior with high values in regions when the observed data @xmath20 and auxiliary data @xmath23 are similar .
for example , uniform kernels are commonplace in likelihood - free models ( e.g. ) , although alternatives such as epanechnikov and gaussian kernels provide improved efficiency .
the resulting approximation to the true posterior target distribution @xmath24\ ] ] improves as @xmath22 decreases , and exactly recovers the target posterior as @xmath25 , as then @xmath26 becomes a point mass at @xmath27 .
posterior simulation from @xmath28 can then proceed via standard simulation algorithms , replacing pointwise evaluations of @xmath28 with monte carlo estimates through the expectation ( [ eqn : marginal - posterior ] ) , based on draws @xmath29 from the model ( e.g. ) .
alternatively , simulation from the joint posterior @xmath30 is available by contriving to cancel the intractable likelihoods @xmath31 in sample weights or acceptance probabilities . for example , importance sampling from the prior predictive distribution @xmath32 results in an importance weight of @xmath33 , which is free of likelihood terms .
see for a discussion of marginal and joint - space likelihood - free samplers .    in general
, the distribution of @xmath31 will be diffuse , unless @xmath23 is discrete and @xmath34 is small . hence , generating @xmath35 with @xmath36 is improbable for realistic datasets @xmath20 , and as a result the degree of computation required for a good likelihood - free approximation @xmath37 ( i.e. with small @xmath22 ) will be prohibitive . in practice , the function @xmath21 is expressed through low dimensional vectors of summary statistics , @xmath38 , such that @xmath21 weights the intractable posterior through ( [ eqn : joint - posterior ] ) with high values in regions where @xmath39 .
if @xmath38 is sufficient for @xmath17 , then letting @xmath25 recovers @xmath40 as before , but with more acceptable computational overheads , as @xmath41 . as sufficient summary statistics are generally unavailable ,
the use of non - sufficient statistics is commonplace .
the effect of less efficient estimators of @xmath17 in ( [ eqn : marginal - posterior ] ) is a more diffuse approximation of @xmath42 .
hence the choice of summary statistics in any application is critical , with the ideal being low - dimensional , efficient and near - sufficient .    in this article
, we implement the likelihood - free sequential monte carlo sampler of , detailed in appendix a. as the class of particle - based algorithms is the most efficient currently available in likelihood - free computation ( e.g. ) , and within this class , the sampler of is the only one to allow non - uniform functions @xmath21 , this sampler provides the best combination of efficient simulation and flexible modelling .
we now develop univariate and multivariate bayesian @xmath0-stable models . unlike existing methods ,
likelihood - free inference is independent of model parameterization .
denote the characteristic function of @xmath11 i.i.d .
univariate @xmath0-stable distributed random variables @xmath43 by @xmath44 a popular and convenient parameterization is @xmath45 \right ) & \text { \ if } \alpha \neq1\\ \exp\left (   i\delta t-\gamma\left\vert t\right\vert \left [ 1+i\beta\frac { 2}{\pi}\text{sgn}\left (   t\right )   \ln\left ( \gamma\left\vert t\right\vert \right ) \right ]   \right ) & \text { \ if } \alpha=1 , \end{array } \right.\ ] ] where @xmath46 and @xmath47 ( e.g. ) .
many alternative parameterizations are detailed in and . under ( [ alphastablecharacteristicfn ] ) ,
the intractable stable density function is continuous and unimodal , taking support on @xmath48 if @xmath49 ; @xmath50 if @xmath51 and @xmath52 otherwise .
efficient simulation of auxiliary data , @xmath16 , under the model is critical for the performance of likelihood - free methods ( section [ sec : abc ] ) . here , it is straightforward to generate @xmath0-stable variates under the model defined by the characteristic function ( [ alphastablecharacteristicfn ] ) ( e.g. ; ) .
this approach is provided in appendix b.      a key component of likelihood - free inference is the availability of low - dimensional , efficient and near - sufficient summary statistics . since @xmath0-stable models can possess infinite variance ( @xmath53 ) and infinite mean ( @xmath54 ) , this choice must be made with care .
here we present several candidate summary vectors , @xmath55@xmath56 , previously utilized for parameter estimation in the univariate @xmath0-stable model . in section [ sec : examples ] we evaluate the performance of these vectors , and provide informed recommendations for the choice of summary statistics under the likelihood - free framework .    0.5 cm _
mcculloch s quantiles _ + and estimate model parameters based on sample quantiles , while correcting for estimator skewness due to the evaluation of @xmath57 , the @xmath58 quantile of @xmath23 , with a finite sample . here
, the data @xmath59 are arranged in ascending order and matched with @xmath60 , where @xmath61 .
linear interpolation to @xmath62 from the two adjacent @xmath63 values then establishes @xmath57 as a consistent estimator of the true quantiles .
inversion of the functions @xmath64 then provides estimates of @xmath65 and @xmath66 . note that from a computational perspective , inversion of @xmath67 or @xmath68 is not required under likelihood - free methods .
finally , we estimate @xmath9 by the sample mean @xmath69 ( when @xmath70 ) .
hence @xmath71    0.5 cm _
zolotarev s transformation _ + based on a transformation of data from the @xmath0-stable family @xmath72 , ( p.16 ) provides an alternative parameterization of the @xmath0-stable model @xmath73 with a characteristic function of the form @xmath74 + \mathbb{c}\left (   \nu^{-\frac{1}{2}}-1\right )   \right\},\ ] ] where @xmath75 is euler s constant , and where @xmath76 , @xmath77 and @xmath78 this parameterization has the advantage that logarithmic moments have simple expressions in terms of parameters to be estimated . for a fixed constant @xmath79 ( recommends @xmath80 ) and for integer @xmath81 , the transformation is @xmath82 defining @xmath83 and @xmath84 , estimates for @xmath85 and @xmath86 are then given by @xmath87 where @xmath88 , using sample variances @xmath89 and @xmath90 .
as before , @xmath9 is estimated by @xmath69 ( for @xmath53 ) , and so @xmath91 .    0.5 cm _ press s method of moments _ + for @xmath92 and unique evaluation points @xmath93 , the method of moments equations obtained from @xmath94 can be solved to obtain (; )
@xmath95 where @xmath96 $ ] .
we adopt the evaluation points @xmath97 and @xmath98 as recommended by , and accordingly obtain @xmath99    0.5 cm _ empirical characteristic function _ + the empirical characteristic function , @xmath100 for @xmath101 , can be used as the basis for summary statistics when standard statistics are not available .
e.g. this may occur through the non - existence of moment generating functions .
hence , we specify @xmath102 where @xmath103 .      ,
@xmath104 , @xmath66 and @xmath9 under the univariate @xmath0-stable model and likelihood - free sampler ( summary statistics @xmath55 ) , based on 10 sampler replicates .
traces are shown as a function ( @xmath23-axis ) of sampler progression ( @xmath105 ) and scale parameter reduction @xmath106 .
parameter values used for data generation are @xmath107 , @xmath108 , @xmath109 and @xmath110 . ]
spectral mass angles ( angle ) and weights ( radius ) for bivariate @xmath0-stable @xmath111 model , with @xmath107 , @xmath112 and @xmath113 . plots
( a)(d ) demonstrate evolution of the estimates for decreasing scale parameter values @xmath22 , based on 10 sampler replicates . ]     discrete spectral masses , @xmath114 , for the trivariate @xmath0-stable @xmath115 model .
true values of the first spectral mass are @xmath116 ( 70% ) and @xmath117 .
point shading indicates mmse value of @xmath118 as a percentage .
the plots demonstrate the evolution of the estimates for decreasing scale parameter values @xmath22 , based on 200 sampler replicates . ]
posterior credibility intervals , based on @xmath119 discrete spectral masses , 20 randomly placed projection vectors and 10 replicate samplers .
broken lines denote estimate of spectral mass using j. p. nolan s _ mvstable _ software , available at academic2.american.edu/@xmath120jpnolan , with ( dashed line ) 40 deterministic projection locations and ( dash - dot line ) 80 projection locations . ]
@xmath0-stable distributions are utilised as models for heavy - tailed noise in many areas of statistics , finance and signal processing engineering . 
 however , in general , neither univariate nor multivariate @xmath0-stable models admit closed form densities which can be evaluated pointwise . 
 this complicates the inferential procedure . as a result , 
 @xmath0-stable models are practically limited to the univariate setting under the bayesian paradigm , and to bivariate models under the classical framework . in this article 
 we develop a novel bayesian approach to modelling univariate and multivariate @xmath0-stable distributions based on recent advances in `` likelihood - free '' inference . 
 we present an evaluation of the performance of this procedure in 1 , 2 and 3 dimensions , and provide an analysis of real daily currency exchange rate data . 
 the proposed approach provides a feasible inferential methodology at a moderate computational cost .    
 * keywords : * @xmath0-stable distributions ; approximate bayesian computation ; likelihood - free inference ; sequential monte carlo samplers .