Bayesian inference for α-stable distributions: A random walk MCMC approach

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Abstract

A novel approach for Bayesian inference in the setting of α-stable distributions is introduced. The proposed approach resorts to a FFT of the characteristic function in order to approximate the likelihood function. The posterior distributions of the parameters are then produced via a random walk MCMC method. Contrary to the existing MCMC schemes, the proposed approach does not require auxiliary variables, and so it is less computationally expensive, especially when large sample sizes are involved. A simulation exercise highlights the empirical properties of the sampler. An application on audio noise data demonstrates how this estimation scheme performs in practical applications.

Introduction

The α-stable family of distributions stems from a more general version of the central limit theorem which replaces the assumption of the finiteness of the variance with a much less restrictive one concerning the regular behavior of the tails (Gnedenko and Kolmogorov, 1954); the Gaussian distribution is thus a particular case of α-stable distribution. This family of distributions has a very interesting pattern of shapes, allowing for asymmetry and thick tails, that makes them suitable for the modelling of several phenomena, ranging from engineering (noise of degraded audio sources) to finance (asset returns); moreover, it is closed under linear combinations.

The family is identified by means of the characteristic functionφ1(t)=expiδ1t-γα|t|α1-iβsgn(t)tanπα2ifα1,expiδ1t-γ|t|1+iβ2πsgn(t)ln|t|ifα=1,which depends on four parameters: α(0,2], measuring the tail thickness (thicker tails for smaller values of the parameter), β[-1,1] determining the degree and sign of asymmetry, γ>0 (scale) and δ1R (location). The shorthand notation I will employ for the distribution is S1α,β,γ,δ1.

While the characteristic function (1) has a quite manageable expression and can straightforwardly produce several interesting analytic results (Zolotarev, 1986), it has unfortunately a major drawback in what concerns estimation and inferential purposes: it is not continuous with respect to the parameters, having a pole at α=1.

An alternative way to write the characteristic function that overcomes this problem, due to Zolotarev (1986), is the following:φ0(t)=expiδ0t-γα|t|α1+iβtanπα2sgn(t)|γt|1-α-1ifα1,expiδ0t-γ|t|1+iβ2πsgn(t)ln(γ|t|)ifα=1.In this case, the distribution will be denoted as S0α,β,γ,δ0. The formulation of the characteristic function is, in this case, quite more cumbersome, and the analytic properties have a less intuitive meaning. At any rate, this formulation is much more useful in what concerns statistical purposes. The only parameter that needs to be “translated” according to the following relationship is δ:δ0=δ1+βγtanπα2ifα1,δ1+β2πγlnγifα=1.On the basis of the above equations, a S1(α,β,1,0) distribution corresponds to a S0(α,β,1,βγtan(πα/2)), provided that α1.

Another parameterization which is sometimes used is the following (Zolotarev, 1986):φ2(t)=expiδ1t-γ2α|t|αexp-iπβ22sgn(t)min(α,2-α)ifα1,expiδ1t-γ2|t|1+iβ22πsgn(t)lnγ2|t|ifα=1.Also in this case, however, the function is not continuous with respect to α and presents a pole at α=1. Another unpleasant feature of this way of writing the characteristic function is that the meaning of the asymmetry parameter β changes according to the value of α: when α(0,1) a negative β indicates negative skewness, whereas for α(1,2) it produces positive skewness. For what concerns the “translation” of this parameterization into (1), one has, for α1: β=cotπα2tanπβ22min(α,2-α),andγ=γ2cosπβ22min(α,2-α)1/α,while δ and α remain unchanged.

Unfortunately, (1), (2) and (4) cannot be inverted to yield a closed-form density function except for a very few cases: α=2, corresponding to the normal distribution, α=1 and β=0, yielding the Cauchy distribution, and α=12,β=±1 for the Lévy distribution.

This difficulty, coupled with the fact that moments of order greater than α do not exist whenever α<2, has made impossible the use of standard estimation methods such as maximum likelihood and the method of moments. Researchers have thus proposed alternative estimation schemes, mainly based on quantiles (McCulloch, 1986). Albeit consistent, however they are not efficient, and most of all they are designed only for the estimation of the distribution parameters: employing them for the estimation of more complex models requires therefore a two-step estimation approach. With the availability of powerful computing machines, it has become possible to exploit computationally-intensive estimation methods for the estimation of α-stable distributions parameters, such as maximum likelihood based on the FFT of the characteristic function (Mittnik et al., 1999) or numerical quadrature of the characteristic function (Nolan, 1997). Those methods, however, present some inconveniences: their accuracy is quite poor for small values of α because of the spikedness of the density function; furthermore, when the parameters are near their boundary, the distributions of the estimators become degenerate making frequentist inferential procedures unreliable.

The Bayesian approach has suffered from the same difficulties as the frequentist one, as the absence of a closed-form density prevented from evaluating the likelihood function and thus constructing posterior inferential schemes. Also in this case, however, the availability of fast computing machines has made possible the use of MCMC methods. In particular, Buckle (1995) has shown that, conditionally on an auxiliary variable, it is possible to express the density function in a closed form. With this result, he proposes a Gibbs sampling scheme for the α-stable distribution parameters.

This approach, however, presents some difficulties. It is unfortunately not straightforward to produce random numbers from this auxiliary variable and one must resort to rejection sampling. Since one needs a random sample from the auxiliary variable of the same size as the observation vector for each iteration of the chain, it follows that this approach can be particularly slow, especially when large sample sizes are involved. Furthermore, since the parameterization involved is (4) and has poles at α=1 and β=0, one needs to a priori constrain α and β on a portion of their support.

Section snippets

A random walk Metropolis sampler

I shall here introduce a novel approach for the construction of the posterior density of the (possibly) asymmetric α-stable law parameters that avoids the use of the auxiliary vector.

To put it on more formal grounds, one looks for a computable expression for the likelihood function in order to be able to produce samples from the posterior distribution of the parameters according to Bayes’ theorem, namely p(α,β,γ,δ|x)p(x|α,β,γ,δ)p(α,β,γ,δ).As I have previously noted, an approximate version of

A practical example

As an illustration on how the proposed method works on real-world data, I have carried out the estimation of a sample of audio noise drawn from a set of recordings of songs taken by Robert Lachmann in Palestine in early twentieth century by means of a mobile recording studio (Lachmann, 1929); as one might guess, the audio medium is very degraded and the noise is extremely heavy tailed. The audio sample consists of 44487 observations and is the same used in Lombardi and Godsill (2006), to which

Concluding remarks

In this paper, I have presented a random walk Metropolis MCMC scheme for the parameters of α-stable distributions. Although it is based on an approximated version of the likelihood, this approach was shown to perform remarkably well, being as twice as fast than the Gibbs sampler proposed by Buckle (1995). This contribution is a very first step in what appears to be a very promising direction; future research will aim at extending this approach to regression and time series models.

In this paper,

Acknowledgements

I would like to thank Steve Brooks, Fabio Corradi and Federico M. Stefanini for their useful comments and especially my co-supervisor Fabrizia Mealli for her insightful suggestions and discussions. A preliminary version of this paper was presented at the SIS 2004 scientific meeting in Bari and at the S.Co. 2005 Conference in Bressanone.

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