On point estimators for Gamma and Beta distributions
Abstract
Let $X_1,\ldots,X_n$ be a random sample from the Gamma distribution with density $f(x)=\lambda^{\alpha}x^{\alpha1}e^{\lambda x}/\Gamma(\alpha)$, $x>0$, 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 $n\geq 4$; moreover, it has finite variance if and only if $n\geq 6$. 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 twoparameter Beta distribution is also given. This is based on a Steintype covariance identity for the Beta distribution, followed by an application of the theory of $U$statistics and the deltamethod. MSC: Primary 62F10; 62F12; Secondary 62E15. Key words and phrases: unbiased estimation; Gamma distribution; Beta distribution; YeChentype closedform estimators; asymptotic efficiency; $U$statistics; Steintype covariance identity; deltamethod.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.10799
 Bibcode:
 2022arXiv220510799P
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Methodology;
 Primary 62F10;
 62F12;
 Secondary 62E15
 EPrint:
 Dedicated to Professor Stavros Kourouklis (18 pages, including one Table)