Power Approximations to Multinomial Tests of Fit

Abstract

Abstract Multinomial tests for the fit of iid observations X 1 …, Xn to a specified distribution F are based on the counts Ni of observations falling in k cells E 1, …, Ek that partition the range of the X j . The earliest such test is based on the Pearson (1900) chi-squared statistic: X 2 = Σ k i=1 (Ni – npi )2/npi , where pi = PF (Xj in Ei ) are the cell probabilities under the null hypothesis. A common competing test is the likelihood ratio test based on LR = 2 Σ k i=1 Ni log(Ni/npi ). Cressie and Read (1984) introduced a class of multinomial goodness-of-fit statistics, R λ, based on measures of the divergence between discrete distributions. This class includes both X 2 (when λ = 1) and LR (when λ = 0). All of the R λ have the same chi-squared limiting null distribution. The power of the commonly used members of the class is usually approximated from a noncentral chi-squared distribution that is also the same for all λ. We propose new approximations to the power that vary with the statistic chosen. Bot...