  In this paper, decision theory was used to derive Bayes and minimax decision
rules to estimate allelic frequencies and to explore their admissibility.
Decision rules with uniformly smallest risk usually do not exist and one
approach to solve this problem is to use the Bayes principle and the minimax
principle to find decision rules satisfying some general optimality criterion
based on their risk functions. Two cases were considered, the simpler case of
biallelic loci and the more complex case of multiallelic loci. For each locus,
the sampling model was a multinomial distribution and the prior was a Beta
(biallelic case) or a Dirichlet (multiallelic case) distribution. Three loss
functions were considered: squared error loss (SEL), Kulback-Leibler loss (KLL)
and quadratic error loss (QEL). Bayes estimators were derived under these three
loss functions and were subsequently used to find minimax estimators using
results from decision theory. The Bayes estimators obtained from SEL and KLL
turned out to be the same. Under certain conditions, the Bayes estimator
derived from QEL led to an admissible minimax estimator (which was also equal
to the maximum likelihood estimator). The SEL also allowed finding admissible
minimax estimators. Some estimators had uniformly smaller variance than the MLE
and under suitable conditions the remaining estimators also satisfied this
property. In addition to their statistical properties, the estimators derived
here allow variation in allelic frequencies, which is closer to the reality of
finite populations exposed to evolutionary forces.
