On estimating the frequency of a recessive gene in a random mating population

The maximum likelihood estimator of the frequency of a recessive gene in a random mating population is simply the square root of the proportion of recessive individuals in a random sample. However, its asymptotic variance, i.e. the inverse of Fisher's information, has an unexpected functional form, and its use may lead to incorrect inferences. An explanation for this is sought by deriving an expansion for the exact variance of the estimator. Further, a number of calculations reveal conditions under which the asymptotic variance provides a reasonable approximation to the exact variance. Lastly, the problem of setting confidence limits to the gene frequency is discussed, with a number of approximations being considered.