Statistics Papers

Document Type

Journal Article

Date of this Version

1988

Publication Source

The Annals of Statistics

Volume

16

Issue

4

Start Page

1567

Last Page

1593

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

L1(F,F^) = ∫[(F(t)−F^(t))2/(F(t)(1−F(t)))] dF(t).

Ordinary Cramer-von Mises loss--L2(F,F^)=∫[(F(t)−F^(t))2]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.

Keywords

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

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Date Posted: 27 November 2017

This document has been peer reviewed.