Asymmetric Distributions from Constrained Mixtures
This provides a method for creating asymmetric distributions for statistical modeling, which is incremental as it builds on existing symmetric distributions like Laplace and normal.
The paper tackles the problem of modeling asymmetric continuous distributions by introducing constrained mixtures, where each mixture component has a shape similar to the base distribution but disjoint domains. The result is the creation of generalized asymmetric versions of Laplace and normal distributions, which are shown to define exponential families with known conjugate priors and closed-form maximum likelihood estimates, and in applications like linear regression and a stock index time-series problem, the asymmetric normal distribution performs at least as well as or better than the symmetric version, with higher likelihood and lower entropy in learned models.
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.