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December, 1988 Admissibility in Discrete and Continuous Invariant Nonparametric Estimation Problems and in their Multinomial Analogs
Lawrence D. Brown
Ann. Statist. 16(4): 1567-1593 (December, 1988). DOI: 10.1214/aos/1176351054

Abstract

Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function $L_1(F, \hat{F}) = \int \big\lbrack (F(t) - \hat{F}(t))^2/(F(t)(1 - F(t))) \big\rbrack dF(t).$ Ordinary Cramer-von Mises loss--$L_2(F, \hat{F}) = \int \lbrack (F(t) - \hat{F}(t))^2 \rbrack dF(t)$--is also studied. Admissibility of the best invariant estimator is investigated. (It is well known in the classical problem that the sample c.d.f. is not the best invariant estimator, and hence is not admissible.) In most discrete settings this estimator must be modified in an obvious fashion to take into account the end points of the known domain of definition for the sample c.d.f. When this is done the resulting estimator is shown to be admissible in some of the discrete settings. However, in the classical continuous setting and in other discrete settings, the best invariant estimator, or its modification, is shown to be inadmissible. Kolmogorov-Smirnov loss for estimating the population c.d.f. is also investigated, but definitive admissibility results are obtained only for discrete problems with sample size 1. In discrete settings the sample median is an admissible estimator of the population median under invariant loss. In the continuous setting this is not true for even sample sizes.

Citation

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Lawrence D. Brown. "Admissibility in Discrete and Continuous Invariant Nonparametric Estimation Problems and in their Multinomial Analogs." Ann. Statist. 16 (4) 1567 - 1593, December, 1988. https://doi.org/10.1214/aos/1176351054

Information

Published: December, 1988
First available in Project Euclid: 12 April 2007

zbMATH: 0684.62017
MathSciNet: MR964939
Digital Object Identifier: 10.1214/aos/1176351054

Subjects:
Primary: 62C15
Secondary: 62D05

Keywords: Admissibility , Cramer-von Mises loss , Kolmogorov-Smirnov loss , multinomial distribution , nonparametric estimation , sample distribution function , sample median

Rights: Copyright © 1988 Institute of Mathematical Statistics

Vol.16 • No. 4 • December, 1988
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