Pearson’s Chi-Square

In this section we derive the asymptotic distribution of the Pearson χ ² statistic as another application of the theorems of Slutsky. We first present three general lemmas relating quadratic forms in normal or asymptotically normal variables to the chi-square distribution. After describing multinomial experiments and the Pearson χ ² statistic for testing a simple null hypothesis, we present two derivations of the asymptotic distribution of Pearson’s χ ² under the null hypothesis. The first, contained in the proof of Theorem 9, is based on the matrix theory notions of rank and projection. The second uses the fact that Pearson’s χ ² is just a version of Hotelling’s T ² and is postponed to the exercises (Exercise 3). We also mention two important variations of Pearson’s χ ², one based on transformations (Hellinger’s χ²) and the other based on the principle of modification (Neyman’s χ ² of Exercise 1).