On Point Estimators for Gamma and Beta Distributions

Let X-1,& mldr;,X-n be a random sample from the Gamma distribution with density f(x)=lambda(alpha)x(alpha-1)e(-lambda x)/Gamma(alpha), x>0, where both alpha>0 (the shape parameter) and lambda>0 (the reciprocal scale parameter) are unknown. The main result shows that the uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter, alpha, exists if and only if n >= 4; moreover, it has finite variance if and only if n >= 6. More precisely, the form of the UMVUE is given for all parametric functions alpha, lambda, 1/alpha and 1/lambda. Furthermore, a highly efficient estimating procedure for the two-parameter Beta distribution is also given. This is based on a Stein-type covariance identity for the Beta distribution, followed by an application of the theory of U-statistics and the delta-method.