## The Annals of Statistics

- Ann. Statist.
- Volume 16, Number 4 (1988), 1567-1593.

### Admissibility in Discrete and Continuous Invariant Nonparametric Estimation Problems and in their Multinomial Analogs

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 16, Number 4 (1988), 1567-1593.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176351054

**Digital Object Identifier**

doi:10.1214/aos/1176351054

**Mathematical Reviews number (MathSciNet)**

MR964939

**Zentralblatt MATH identifier**

0684.62017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C15: Admissibility

Secondary: 62D05: Sampling theory, sample surveys

**Keywords**

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

#### Citation

Brown, Lawrence D. Admissibility in Discrete and Continuous Invariant Nonparametric Estimation Problems and in their Multinomial Analogs. Ann. Statist. 16 (1988), no. 4, 1567--1593. doi:10.1214/aos/1176351054. https://projecteuclid.org/euclid.aos/1176351054