On Point Estimators for Gamma and Beta Distributions

Let X1,…,Xn be a random sample from the gamma distribution with density f(x)=λαxα−1e−λx/Γ(α), x > 0, where both α>0 (the shape parameter) and λ>0 (the reciprocal scale parameter) are unknown. The main result shows that the uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter, α, 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 α, λ, 1/α, and 1/λ. 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.