On Point Estimators for Gamma and Beta Distributions

Abstract

Let X 1 , … , X n be a random sample from the gamma distribution with density f ( x ) = λ α x α − 1 e − λ x / 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.