On point estimators for Gamma and Beta distributions
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Let X_1,…,X_n be a random sample from the Gamma distribution with density f(x)=λ^αx^α-1e^-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. MSC: Primary 62F10; 62F12; Secondary 62E15. Key words and phrases: unbiased estimation; Gamma distribution; Beta distribution; Ye-Chen-type closed-form estimators; asymptotic efficiency; U-statistics; Stein-type covariance identity; delta-method.
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