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The Best Estimator and a Strongly Consistent Asymptotically Normal Unbiased Estimator of Lorenz Curve Gini Index and Theil Entropy Index of Pareto Distribution

T. S. K. Moothathu
Sankhyā: The Indian Journal of Statistics, Series B (1960-2002)
Vol. 52, No. 1 (Apr., 1990), pp. 115-127
Published by: Springer on behalf of the Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25052627
Page Count: 13
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The Best Estimator and a Strongly Consistent Asymptotically Normal Unbiased Estimator of Lorenz Curve Gini Index and Theil Entropy Index of Pareto Distribution
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Abstract

This paper presents the uniformly minimum variance unbiased estimator and a strongly consistent, asymptotically normal, unbiased estimator of each of the Lorenz curve L(p, a), the Gini index λ(a), and the Theil entropy index η(a) of a two-parameter Pareto distribution in the case when the shape parameter a alone is unknown and in the case when both a and the scale parameter θ are unknown. Further the variance of each estimator is derived. The estimator of L(p, a) is in terms of the Bessel function ${}_{0}F{}_{1}$, and its variance is in terms of $\psi _{2}$, the confluent hypergeometric function of two variables. The estimator of λ(a) is in terms of Kummer's function ${}_{1}F{}_{1}$ and its variance is in terms of $F_{2}$, the Appel function of the second kind. The estimator of η(a) is in terms of the generalized hypergeomtric function ${}_{2}F{}_{2}$ and its variance is in terms of the Kempé-de-Fériet function, which is a generalization of both $F_{2}$ and $\psi _{2}$.

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