Multinomial Goodness-Of-Fit Tests

Abstract

SUMMARY This article investigates the family {I λ;λ ϵ ℝ} of power divergence statistics for testing the fit of observed frequencies {Xi; i = 1, …, k} to expected frequencies {Ei; i = 1, …, k}. From the definition 2nIλ=2λ(λ+1)∑i=1kXi{(XiEi)λ−1};λ∈ℝ it can easily be seen that Pearson's X 2 (λ = 1), the log likelihood ratio statistic (λ = 0), the Freeman-Tukey statistic (λ = –½) the modified log likelihood ratio statistic (λ = –1) and the Neyman modified X 2 (λ = –2), are all special cases. Most of the work presented is devoted to an analytic study of the asymptotic difference between different I λ however finite sample results have been presented as a check and a supplement to our conclusions. A new goodness-of-fit statistic, where λ = ⅔, emerges as an excellent and compromising alternative to the old warriors, I 0 and I 1.