The distribution of Pearson residuals in generalized linear models

TL;DR

This paper derives an asymptotic formula for the distribution of Pearson residuals in continuous generalized linear models, enabling more accurate residual analysis through corrected residuals that match the true distribution to order n^{-1}.

Contribution

It introduces corrected Pearson residuals with an asymptotic density formula, improving residual diagnostics in generalized linear models.

Findings

Corrected residuals match true residual distribution to order n^{-1}.

Simulation results demonstrate improved residual approximation.

Applications provided for key GLM types.

Abstract

In general, the distribution of residuals cannot be obtained explicitly. We give an asymptotic formula for the density of Pearson residuals in continuous generalized linear models corrected to order $n^{-1}$, where $n$ is the sample size. We define corrected Pearson residuals for these models that, to this order of approximation, have exactly the same distribution of the true Pearson residuals. Applications for important generalized linear models are provided and simulation results for a gamma model illustrate the usefulness of the corrected Pearson residuals.