Skewness of maximum likelihood estimators in dispersion models

TL;DR

This paper derives formulas for the skewness of maximum likelihood estimators in dispersion models, extending previous results and demonstrating their practical importance through simulations.

Contribution

It introduces a matrix formula for the skewness of MLEs in dispersion models, applicable to various submodels and parameters, extending prior work.

Findings

Derived skewness formulas for regression, precision, and dispersion parameters.

Formulas are suitable for computer implementation and application.

Simulation study confirms practical relevance of the results.

Abstract

We introduce the dispersion models with a regression structure to extend the generalized linear models, the exponential family nonlinear models (Cordeiro and Paula, 1989) and the proper dispersion models (J{\o}rgensen, 1997a). We provide a matrix expression for the skewness of the maximum likelihood estimators of the regression parameters in dispersion models. The formula is suitable for computer implementation and can be applied for several important submodels discussed in the literature. Expressions for the skewness of the maximum likelihood estimators of the precision and dispersion parameters are also derived. In particular, our results extend previous formulas obtained by Cordeiro and Cordeiro (2001) and Cavalcanti et al. (2009). A simulation study is perfomed to show the practice importance of our results.