Estimating Allele Frequencies and Inbreeding Coefficients in K-Allele Models

Some of the methods of estimation of allele frequencies and inbreeding coefficients in a K-allele model are examined. A result that has long been assumed to be true is proved. That is, in the presence of inbreeding, the maximum likelihood estimators of the allele frequencies and of the inbreeding coefficient f do not in general equal their observed (or sample) values (except when K 1⁄4 2). A least-squares way of looking at the estimation problem is presented, and simulations are used to compare the three types of estimators (sample, maximum likelihood, and leastsquares) in a 3-allele model. Probability generating functions are used to derive exact expressions for the bias of the sample estimator of f in a 2-allele model for any sample size, and those biases are calculated for a number of situations. Finally, an approximately unbiased estimator of the inbreeding coefficient when an allele is rare or common is proposed, and its bias is compared with that of the sample estimator and with that of an estimator proposed by Weir (1996).