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September, 1989 Inadmissibility of the Empirical Distribution Function in Continuous Invariant Problems
Qiqing Yu
Ann. Statist. 17(3): 1347-1359 (September, 1989). DOI: 10.1214/aos/1176347274

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.

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Qiqing Yu. "Inadmissibility of the Empirical Distribution Function in Continuous Invariant Problems." Ann. Statist. 17 (3) 1347 - 1359, September, 1989. https://doi.org/10.1214/aos/1176347274

Information

Published: September, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0691.62012
MathSciNet: MR1015156
Digital Object Identifier: 10.1214/aos/1176347274

Subjects:
Primary: 62C15
Secondary: 62D05

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 3 • September, 1989
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