2022/05/22 by Nickos Papadatos, Papadatos, Nickos
Computer Science · Mathematics · #62F12 #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Point processes and geometric inequalities #Primary 62F10 #Secondary 62E15 #Statistical Distribution Estimation and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2205.10799
openalex publication_date 2022/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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.