Power Approximations to Multinomial Tests of Fit

Abstract

Multinomial tests for the fit of iid observations X ₁ …, X_n to a specified distribution F are based on the counts N_i of observations falling in k cells E ₁, …, E_k 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 (N_i – np_i )²/np_i , where p_i = P_F (X_j in E_i ) 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 N_i log(N_i/np_i ). 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. Both the computation and results on asymptotic error rates suggest that the new approximations are greatly superior to the traditional power approximation for statistics R ^λother than the Pearson X ². The derivation of the limiting null distribution for the Cressie—Read statistics, following that for LR, is based on a Taylor series expansion of R ^λ, in which X ² is the dominant term. The same expansion produces the traditional noncentral chi-squared power approximation by considering sequences of alternative distributions for the X_j that approach the hypothesis F at a suitable rate. Our power approximations are obtained from a Taylor series expansion that is valid for arbitrary sequences of alternatives. When linear and quadratic terms are retained, an accurate but computationally difficult approximation, A ^λ, in terms of linear combinations of noncentral chi-squares is obtained. A second approximation, B ^λ, in terms of a single noncentral chi-squared distribution results from averaging the coefficients in A ^λ, This simple approximation performs well. In the important case of the statistic LR, A ^λ= B ^λand this new noncentral chi-squared approximation is very accurate. Retaining only linear terms in the expansion produces an approximation L ^λbased on a normal distribution; this is generally much inferior to A ^λand B^λ.

Key Words:

• Asymptotic expansions
• Goodness of fit
• Noncentral chi-squared distribution