Efficient estimators for the good family |
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Authors: | Louis G Doray Andrew Luong |
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Institution: | 1. Département de mathématiques et de statistique , Université de Montréal , Montréal, Québec, H3C 3J7, CanadaC.P. 6128, Succursale Centre-ville;2. école d'actuariat , Université Laval , Ste-Foy, Quebéc, G1K 7P4, CanadaCite Universitaire |
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Abstract: | We consider the problem of estimating the two parameters of the discrete Good distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. The maximum likelihood estimators (MLE's) of the parameters are obtained by solving numerically a system of equations involving the Lerch zeta function and the sufficient statistics. We find an expression for the asymptotic variance-covariance matrix of the MLE's, which can be evaluated numerically. We show that the probability mass function satisfies a simple recurrence equation linear in the two parameters, and propose the quadratic distance estimator (QDE) which can be computed with an ineratively reweighted least-squares algorithm. the QDE is easy to calculate and admits a simple expression for its asymptotic variance-covariance matrix. We compute this matrix for the MLE's and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. Finally, we present a numerical example. |
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Keywords: | quadratic distance estimator iteratively reweighted least-squares maximum likelihood Lerch zeta function asymptotic efficiency zeta distribution Lerch distribution |
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