Approximating the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit

Abstract

Abstract Approximations were derived for the mean and variance of G 2, the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution. These approximate moments, accurate to O(N -3), may be used in fitting the distribution of G 2. Extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability. The asymptotic chi-squared distribution was found to fit poorly for k ≥ 4. Satisfactory results were obtained using a multiplicative correction to the chi-squared fit. More sophisticated procedures using the beta distribution produced increased accuracy, but at the cost of excessive computational labor.