Some new Laplace-based probability distributions for modeling data with pronounced peaks combined with heavy tails and outliers
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Authors
Natido, Amos
Issue Date
2024
Type
Dissertation
Language
en_US
Keywords
Asymmetric Distributions , EM-Algorithm , Generalized Laplace Distribution , Markov Chain Monte Carlo , Normal Variance-Mean Mixture Models , Uniform Distribution
Alternative Title
Abstract
Since the publication of Kotz et al. (2001), the classical Laplace distribution and its skewed generalizations have become increasingly important in modeling univariate and multivariate data across diverse application areas. In this study, we explore several extensions of the Laplace model, both univariate and multivariate, which are useful for modeling either symmetric or asymmetric data with pronounced peaks combined with long and heavy tails. First, we revisit the class of univariate uniform-Laplace mixture (ULM) distributions, which are well-suited to model these features with great flexibility. Our primary objective here is to address the estimation issues associated with this model. In turn, the ULM distributions can arise as marginal distributions of a general class of multivariate distributions we consider next, where each coordinate is conditionally Gaussian but flexible enough to allow for different shapes or tail behaviors. We provide a general theory for this construction, focusing particularly on an extension of the multivariate generalized Laplace (GAL) distribution. In this extension, the shape and tail characteristics along each dimension are controlled by a univariate GAL random variable with distinct characteristics. We also examine another type of multivariate distribution of current interest, where the stochastic “mixing" variables are placed along the eigenvectors of the underlying covariance matrix rather than coordinate-wise. We develop theoretical properties of this model and present several examples, including the case of gamma mixing variables. Here, each marginal is associated with a linear combination of GAL random variables. These novel classes of probability distributions offer enhanced flexibility in modeling multidimensional data that may exhibit varying degrees of skewness, tail behavior, and outliers along each dimension. We develop the fundamental properties of these classes of Laplace-based models and discuss parameter estimation through various computational schemes. This includes leveraging the Expectation Maximization (EM) algorithm and its variants, and addressing computational issues related to the models. We investigate the effectiveness of these estimation strategies on simulated data and demonstrate applications to real data examples.