Abstract

Extreme values and skewness are often observed in engineering, financial and biological time-series. This talk summarizes my PhD work, a study motivated by the need of efficient and reliable Bayesian inference methods when the α-stable model is selected to represent such data. While having a key role as the limit of the generalized central limit theorem (CLT), the class of stable distributions is highly intractable, given that it is not possible to analytically express its pdf. Several approximate methods are available in the literature, in both the frequentist and Bayesian paradigms, but they suffer from a number of deficiencies, the most relevant being the lack of quantification of the approximation made.

This talk focuses on two different latent variable models, that provide two marginal representations of the stable pdf. For the first model, an exact parameter inference scheme, based on the pseudo-marginal Markov chain Monte Carlo approach, is developed, providing results comparable to a state of the art Bayesian sampler. The novel method does not introduce any approximation, while allowing for better control of the quality of the inference. The second model derives from an infinite series representation stable random variables. In this setting, we first formulate a CLT for the series residual, which serves to justify existing approximations used in previous literature. Moreover, we present numerical and theoretical results on the rate of convergence for finite values of the series truncation parameter, thus giving theoretical guarantees on the accuracy achieved. Finally, we present extensions of this model to multivariate stable random variables, in the framework of simulation of continuous time stochastic processes. This is at the basis of inference methods to be developed in future work.