Expectations of some ratio-type estimators under the gamma distribution

TL;DR

This paper derives simplified formulas for the expectations of various ratio-type estimators, such as the Gini, Theil, and Atkinson indices, under the gamma distribution, using distributional properties and the gamma-beta relationship.

Contribution

It provides new, simpler proofs for the expected values of common ratio-type estimators under the gamma distribution, enhancing analytical tractability.

Findings

Expected values of Gini, Theil, and Atkinson indices derived

Simpler proofs using gamma distribution properties

Expected variance-to-mean ratio also obtained

Abstract

We study the expectations of some ratio-type estimators under the gamma distribution. Expectations of ratio-type estimators are often difficult to compute due to the nature that they are constructed by combining two separate estimators. With the aid of Lukacs' Theorem and the gamma-beta (gamma-Dirichlet) relationship, we provide alternative proofs for the expected values of some common ratio-type estimators, including the sample Gini index, the sample Theil index, and the sample Atkinson index, under the gamma distribution. Our proofs using the distributional properties of the gamma distribution are much simpler than the existing ones. In addition, we also derive the expected value of the sample variance-to-mean ratio under the gamma distribution.