Normal Goodness-of-Fit Tests for Multinomial Models with Large Degrees of Freedom

Abstract

Goodness-of-fit tests for independent multinomials with parameters to be estimated are usually based on Pearson's X ² or the likelihood ratio G ². Both are included in the family of power-divergence statistics SD _λ. For increasing sample sizes each SD _λ has an asymptotic X ² distribution, provided that the number of cells remains fixed. We are dealing with an increasing-cells approach, where the number J of independent multinomials increases while the number of classes for each multinomial and the number of parameters remain fixed. Extending results on X ² and G ², the asymptotic normality of any SD _λ is obtained for increasing cells. The corresponding normal goodness-of-fit tests discussed here apply for models with large degrees of freedom with no restrictions imposed on the sizes N _j of each multinomial, allowing large as well as small expectations (sparse data) within each cell. The asymptotic expectation and variance of SD _λ are easy to compute for Pearson's X ² and simplify considerably for general λ if the harmonic or arithmetic mean of the multinomial sizes N _j are large. Applications to quantal response models for binomial data are treated in more detail. It turns out that the asymptotic expectation and variance of X ² and G ² agree to first order with the conditional moments given the estimated parameters. And for binary data (with N _j = 1 for all j), the goodness-of-fit tests appear as score tests with respect to an enlarged model. Our presentation focuses on general aspects of the tests and its applications rather than on formal proofs for the underlying limit results, which are only outlined here.

Key Words:

• Binomial data
• Likelihood ratio
• Pearson's chi square
• Power-divergence statistic
• Sparse data