On the Asymptotic Moments of Pearson Type Statistics Based on Resampling Procedures

This paper presents a discussion of overdispersed generalized linear regression models. Commonly used estimates of the unknown dispersion parameter are the Pearson statistic or the modification of Farrington (1996), who proposed a first order linear correction term to Pearson's statistic. We like to focus our attention onto a bootstrap statistic based on resampling from some observed centered Pearson residuals. Especially, if these residuals are also scaled by a constant term the respective bootstrap dispersion corresponds to the empirical variance of the Pearson residuals. Hence, this statistic includes an additional quadratic term. Assuming exponential family properties, this bootstrap dispersion estimate is shown to be asymptotically uncorrelated to the regression parameter estimates. Moreover, approximations to the first two unconditional moments are obtained, which can be easily compared to those of the Pearson statistic