Double asymptotics for the chi-square statistic

TL;DR

This paper investigates the asymptotic behavior of the Pearson chi-square statistic as the number of classes grows with the sample size, revealing different limiting distributions depending on the growth rate.

Contribution

It establishes the distributional limits of the chi-square statistic under varying growth conditions of classes relative to sample size.

Findings

Gaussian limit when n/√m → ∞

Poisson limit when n/√m → λ > 0

Degenerate limit when n/√m → 0

Abstract

We consider distributional limit of the Pearson chi-square statistic when the number of classes m increases with the sample size n in such way that $n/\sqrt{m} \to {\lambda}$. Under mild moment conditions, the limit is Gaussian for ${\lambda} = \infty$, Poisson for finite ${\lambda} > 0$, and degenerate for ${\lambda} = 0$.