 Quadratic Pearson approximation to the Poisson deviance

ID: quadratic-pearson-approximation-to-the-poisson-deviance

Write

y_{i} = e_{i} + r_{i}

. For

∣ r_{i} ∣/ e_{i}

small, the cell contribution to the Poisson deviance expands as

2 [(e_{i} + r_{i}) lo g (1 + r_{i} / e_{i}) - r_{i}] = \frac{r _{i}^{2}}{e _{i}} - \frac{r _{i}^{3}}{3 e _{i}^{2}} + O (r_{i}^{4} / e_{i}^{3}) .

Summing gives the Pearson chi-squared statistic as the quadratic approximation. The approximation can be inaccurate for sparse expected counts or cells with relative residuals near minus one.

 New to topics? Read the docs here!