Improved estimators for dispersion models with dispersion covariates

TL;DR

This paper develops explicit second-order bias correction formulas for estimators in extended dispersion models with covariates, improving estimation accuracy and computational efficiency.

Contribution

It introduces new closed-form second-order bias correction formulas for regression and dispersion parameters in extended dispersion models with covariates.

Findings

Bias-corrected estimators outperform uncorrected ones in simulations.

Formulas simplify bias correction to a weighted linear regression.

Bias corrections are effective for models with closed-form information matrices.

Abstract

In this paper we discuss improved estimators for the regression and the dispersion parameters in an extended class of dispersion models (J{\o}rgensen, 1996). This class extends the regular dispersion models by letting the dispersion parameter vary throughout the observations, and contains the dispersion models as particular case. General formulae for the second-order bias are obtained explicitly in dispersion models with dispersion covariates, which generalize previous results by Botter and Cordeiro (1998), Cordeiro and McCullagh (1991), Cordeiro and Vasconcellos (1999), and Paula (1992). The practical use of the formulae is that we can derive closed-form expressions for the second-order biases of the maximum likelihood estimators of the regression and dispersion parameters when the information matrix has a closed-form. Various expressions for the second-order biases are given for special models. The formulae have advantages for numerical purposes because they require only a supplementary weighted linear regression. We also compare these bias-corrected estimators with two different estimators which are also bias-free to the second-order that are based on bootstrap methods. These estimators are compared by simulation.