Extreme values and skewness in time-series are often observed in engineering, financial
and biological applications. This thesis is a study motivated by the need of efficient
and reliable Bayesian inference methods when the α-stable model is selected to
represent such data.
The class of stable distributions is the limit of the generalized central limit
theorem (CLT), having a key role in representing phenomena that can be thought of
as the sum of many perturbations, with potentially unbounded variance. Besides the
ability to model heavy-tailedness, another consequence of the generalized CLT is a
further degree of freedom of stable distributions, namely their potential skewness.
However, stable distributions are, at the same time, highly intractable for inference
purposes. Several approximate methods are available in the literature, in both the
frequentist and Bayesian paradigms, but they suffer from a number of deficiencies,
the greatest of which is the lack of quantification of the approximation in place. This
thesis proposes Bayesian inference schemes for two different latent variable models,
with the aim of providing guarantees of accuracy when the α-stable model is used.
In the first part of the thesis, a marginal representation of the α-stable density
is used to develop a novel, asymptotically exact, Bayesian method for parameter
inference. This is based on the pseudo-marginal Markov chain Monte Carlo (MCMC)
approach, that requires only unbiased estimates of the intractable likelihood, computed
through adaptive importance sampling for the marginal representation. The
results obtained are comparable to a state of the art conditional Gibbs sampler, but
do not introduce any approximation, while allowing for better control of the quality
of the inference.
The focus of the second and central part of the thesis is the Poisson series
representation (PSR) of α-stable random variables. An approach that turns the
infinite-dimensional PSR into an approximately conditionally Gaussian representation,
by means of Gaussian approximation of the residual of the series, has been presented
in previous literature, together with inference procedures such as MCMC and Particle
Filtering. In this setting, the first contribution of this dissertation is the formulation
of a CLT for the PSR residual, which serves to justify the existing approximation.
Moreover, numerical and theoretical results on the rate of convergence for finite values
of the truncation parameter are presented. The convergence is examined directly in
terms of Kolmogorov distance between distribution functions, through the application
of probability theoretic results, such as the Esseˊen’s smoothing lemma. This analysis
allows for the selection of appropriate truncations for different α-stable parameter
configurations and gives theoretical guarantees on the accuracy achieved when using
the PSR model. Furthermore, superior behaviour of the proposed approximation is
found, compared to the simple series truncation, justifying its use for inference tasks.
In the third and final part of this thesis, an extension of the modified Poisson
series representation (MPSR) of linear continuous-time models driven by α-stable
Leˊvy processes to the multivariate case is presented. Stable Leˊvy processes are
suitable to model jumps and discontinuities in the state, while possessing the self-similarity
property, which makes these processes a very natural class for the driving
noise in continuous time models. A scheme for approximate simulation from the
multivariate linear models, namely multivariate stable vectors evolving in time, is
presented. While stable random vectors are parametrized by a function, the presented
approximate approach involves only finite dimensional parameters. This will facilitate
inference methods, to be developed in future work, towards which the proposed
simulation methods constitute the foundational work