  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.
