Jackknifing and Bootstrapping Goodness-of-Fit Statistics in Sparse Multinomials
Abstract Many goodness-of-fit problems can be put in the following form: n = (n 1, …, nk ) is a multinomial (N, π) vector, and it is desired to test the composite null hypothesis H 0: π = p(θ) against all possible alternatives. The usual tests used are Pearson's (1900) statistic or the likelihood ratio statistic (Neyman and Pearson 1928) . Cressie and Read (1984) pointed out that both of these statistics are in the power family of statistics , with λ = 1 and 0, respectively; they suggested an alternative statistic with . Although all of these statistics are asymptotically χ2 in the usual situation of K fixed and N → ∞, this is not the case if the multinomial is sparse; specifically, Morris (1975) showed that, under certain regularity conditions with K and N → ∞, both X 2 and G 2 are asymptotically normal (with different mean and variance) under a simple null hypothesis. Cressie and Read (1984) extended these results to the general 2NI λ family. Although these results have not been proven for composite nul...