Pearson's Chi-squared Statistic

Definition

The Pearson’s ${\chi }^2$ statistic is goodness-of-fit measure for GLM defined as $X^2=\sum _{i=1}^n\frac{(y_i-{\hat{\mu }}_i{)}^2}{V({\hat{\mu }}_i)}\stackrel{a}{\sim }{\chi }^2(n-p)$ where $\hat{\mu }=\mu (\theta )=b^{\prime }(\theta )$, $V(\hat{\mu })=b^{\prime \prime }(\theta )$ is the variance function, $p$ is the number of parameter of the current model, and $n$ is the sample size.

Facts

Under the Gaussian distribution, the Deviance and the Pearson’s ${\chi }^2$ statistic is the same and follow Chi-squared Distribution $D=X^2\sim {\chi }^2(n-p)$