Asymptotics for a 2 × 2 Table with Fixed Margins

Two coins A and B are tossed N1 and N2 times, respectively. Denote by M1 (M2) the total number of heads (tails) and X the number of heads from coin A. It is well known that conditional on the Mi and Ni, X has 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 approximating X and estimating θ are of major concerns. Based on a connection between the probability generating function of X and the classical Jacobi polynomials, we show that X is 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)] →L N(0, 1) holds if and only if M1M2N1N2/N 3 → ∞, where N = N1+N2. 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 X to be approximated by the Poisson distribution is established as well.