Pearson Residual

Definition

The Pearson residual is defined as $r_{P_i}=\frac{y_i-{\hat{\mu }}_i}{\sqrt{V({\hat{\mu }}_i)}}$ where $\hat{\mu }=\mu (\theta )=b^{\prime }(\theta )$, $V(\hat{\mu })=b^{\prime \prime }(\theta )$ is the variance function.

Facts

$X^2=\sum _{i=1}^n{r}_{P_i}^2$ where $X^2$ is the Pearson’s Chi-squared Statistic