Asymptotic adjustments of Pearson residuals in exponential family nonlinear models

TL;DR

This paper introduces corrected and PCA Pearson residuals for exponential family nonlinear models, improving distributional approximation and correlation properties, with numerical evidence supporting their effectiveness.

Contribution

The paper proposes new corrected and PCA Pearson residuals that better approximate true residuals and are approximately uncorrelated, extending previous work to nonlinear models.

Findings

PCA residuals are approximately normally distributed.

Corrected residuals match the true residuals' distribution up to order O(n^{-1}).

PCA residuals are approximately uncorrelated.

Abstract

In this work we define a set of corrected Pearson residuals for continuous exponential family nonlinear models that have the same distribution as the true Pearson residuals up to order $\mathcal{O}(n^{-1})$, where $n$ is the sample size. Furthermore, we also introduce a new modification of the Pearson residuals, which we call PCA Pearson residuals, that are approximately uncorrelated. These PCA residuals are new even for the generalized linear models. The numerical results show that the PCA residuals are approximately normally distributed, thus improving previous results by Simas and Cordeiro (2009). These numerical results also show that the corrected Pearson residuals approximately follow the same distribution as the true residuals, which is a considerable improvement with respect to the Pearson residuals and also extends the previous work by Cordeiro and Simas (2009).