The Annals of Statistics

Inadmissibility of the Empirical Distribution Function in Continuous Invariant Problems

Qiqing Yu

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Abstract

Consider the classical invariant decision problem of estimating an unknown continuous distribution function $F,$ with the loss function $L(F, a) = \int(F(t) - a(t))^2\lbrack F(t) \rbrack^\alpha \lbrack 1 - F(t) \rbrack^\beta dF(t),$ and a random sample of size $n$ from $F.$ It is proved that the best invariant estimator is inadmissible when: 1. $ n > 0, - 1 < \alpha, \beta \leq 0 \text{and} -1 \leq \alpha + \beta.$ 2. $ n > 0, -1 < \alpha = \beta \leq - \frac{1}{2}.$ 3. $ n > 1, (\mathrm{i}) \alpha = -1 \text{and} \beta = 0, \text{or} (\mathrm{ii}) \alpha = 0 \text{and} \beta = -1.$ 4. $ n > 2, \alpha = \beta = -1.$ Thus the empirical distribution function, which is the best invariant estimator when $\alpha = \beta = -1,$ is inadmissible when $n \geq 3.$ This extends some results of Brown.

Article information

Source
Ann. Statist., Volume 17, Number 3 (1989), 1347-1359.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347274

Digital Object Identifier
doi:10.1214/aos/1176347274

Mathematical Reviews number (MathSciNet)
MR1015156

Zentralblatt MATH identifier
0691.62012

JSTOR
links.jstor.org

Subjects
Primary: 62C15: Admissibility
Secondary: 62D05: Sampling theory, sample surveys

Keywords
Admissibility invariant estimator empirical distribution function nonparametric estimator Cramer-von Mises loss

Citation

Yu, Qiqing. Inadmissibility of the Empirical Distribution Function in Continuous Invariant Problems. Ann. Statist. 17 (1989), no. 3, 1347--1359. doi:10.1214/aos/1176347274. https://projecteuclid.org/euclid.aos/1176347274


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