Simple improvements on Cornfield's approximation to the mean of a noncentral hypergeometric random variable

This note provides some simple improvements to Cornfield's asymptotic approximation to the mean of a noncentral hypergeometric random variable. Simplicity, in the sense of avoidance of exact computation in favour of easy approximations, derives from an observation of Mantel & Hankey that in a four-fold table with all margins fixed, the ratio of expected cross-products is equal to the odds ratio parameter governing the noncentral distribution of the reference cell. The improvements often reduce the relative error of approximation by one or more powers often, even for tables whose total sample size is as small as two.