Expectations of some ratio-type estimators under the gamma distribution
Jia-Han Shih
TL;DR
This paper examines the expectations of ratio-type estimators under the gamma distribution, focusing on the sample Gini, Theil, and Atkinson indices, and the variance-to-mean ratio (VMR). By leveraging Lukacs' Theorem and the gamma-beta (gamma-Dirichlet) relationship, it provides concise alternative proofs for the expected values, avoiding more involved Laplace-transform arguments. The results show that the sample Gini index is unbiased under the gamma distribution ($\mathbb{E}(G_n)=G$), while the Theil and Atkinson indices are biased with explicit formulas for $\mathbb{E}(T_n)$ and $\mathbb{E}(A_n)$, and the sample VMR exhibits downward bias, with $\mathbb{E}(\text{VMR}_n)=\dfrac{n\alpha}{(n\alpha+1)\lambda}$. These distribution-specific insights provide compact, exact tools for assessing ratio-type estimators in gamma-model contexts and may inform inequality analyses under gamma-like assumptions.
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.