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July, 1974 The Asymptotic Sufficiency of a Relatively Small Number of Order Statistics in Tests of Fit
Lionel Weiss
Ann. Statist. 2(4): 795-802 (July, 1974). DOI: 10.1214/aos/1176342766

Abstract

For each $n, X_n(1), \cdots, X_n(n)$ are independent and identically distributed continuous random variables over $(0, 1)$, with common density function equal to $1 + r(x)/n^{\frac{1}{2}}, r(x)$ unknown but satisfying certain regularity conditions. The problem is to test the hypothesis that $r(x) = 0$ for all $x$ in (0, 1). $Y_n(1) < \cdots < Y_n(n)$ are the ordered values of $X_n(1), \cdots, X_n(n). \delta$ is a fixed value in the open interval $(\frac{3}{4}, 1)$. It is shown that $Y_n(\lbrack n^\delta\rbrack), Y_n(2\lbrack n^\delta\rbrack), \cdots$ are asymptotically sufficient, and can be assumed to have a joint normal distribution for all asymptotic purposes. Using these facts, a test of the hypothesis is constructed with a good asymptotic power curve.

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Lionel Weiss. "The Asymptotic Sufficiency of a Relatively Small Number of Order Statistics in Tests of Fit." Ann. Statist. 2 (4) 795 - 802, July, 1974. https://doi.org/10.1214/aos/1176342766

Information

Published: July, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0284.62019
MathSciNet: MR365889
Digital Object Identifier: 10.1214/aos/1176342766

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 4 • July, 1974
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