Asymmetric Distributions from Constrained Mixtures

TL;DR

This paper develops a new class of constrained mixture distributions to create asymmetric variants of common distributions like Laplace and normal, with applications in regression and time-series modeling.

Contribution

It introduces constrained mixtures for continuous distributions, enabling the creation of generalized asymmetric distributions with known properties and closed-form maximum likelihood estimates.

Findings

Asymmetric normal performs at least as well as symmetric in regression.

Asymmetric distributions yield higher likelihood in stock index modeling.

Constrained mixtures facilitate modeling of asymmetric data.

Abstract

This paper introduces constrained mixtures for continuous distributions, characterized by a mixture of distributions where each distribution has a shape similar to the base distribution and disjoint domains. This new concept is used to create generalized asymmetric versions of the Laplace and normal distributions, which are shown to define exponential families, with known conjugate priors, and to have maximum likelihood estimates for the original parameters, with known closed-form expressions. The asymmetric and symmetric normal distributions are compared in a linear regression example, showing that the asymmetric version performs at least as well as the symmetric one, and in a real world time-series problem, where a hidden Markov model is used to fit a stock index, indicating that the asymmetric version provides higher likelihood and may learn distribution models over states and transition distributions with considerably less entropy.