Statistica Sinica

S. G. Kou and Z. Ying

Abstract:Two coinsAandBare tossedNand_{1}Ntimes, respectively. Denote by_{2}M(_{1}M) the total number of heads (tails) and_{2}Xthe number of heads from coinA. It is well known that conditional on theMand_{i}N,_{i}Xhas a noncentral hypergeometric distribution which depends only on the odds ratio θ between the success probabilities of the two coins. This model is commonly used in the analysis of a single 2 × 2 table, in which approximatingXand estimating θ are of major concerns. Based on a connection between the probability generating function ofXand the classical Jacobi polynomials, we show thatxis equal in distribution to a sum of independent, though not identically distributed, Bernoulli random variables. It is then established that the central limit theorem (X-EX)/[Var(X)]^{1/2}→αN(0,1) holds if and only ifM→ ∞, where_{1}M_{2}N_{1}N_{2}/N^{3}N=N. In addition, this minimum condition is shown to be sufficient for (1) the maximum likelihood estimator of θ and the empirical odds ratio to be consistent and asymptotically normal, (2) some classical estimators for the asymptotic variance of the empirical odds ratio, such as those suggested in Cornfield (1956) and Woolf (1955), as well as a new variance estimator to be consistent. A Berry-Esseen-type bound is found, and a necessary and sufficient condition for_{1}+N_{2}Xto be approximated by the Poisson distribution is established as well.

Key words and phrases:Noncentral hypergeometric distribution, empirical odds ratio, generating function, asymptotic normality, maximum likelihood estimator, Jacobi polynomials, Poisson approximation.