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June, 1962 Some Modified Kolmogorov-Smirnov Tests of Approximate Hypotheses and their Properties
Judah Rosenblatt
Ann. Math. Statist. 33(2): 513-524 (June, 1962). DOI: 10.1214/aoms/1177704577


The paradox of almost certain rejection of the null hypothesis in the Chi Square test-of-fit, when many observations are used, has been pointed out by Cochran [4], and largely removed by Lehmann and Hodges [7]. The same paradox arises in most tests of goodness-of-fit. In this paper the Kolmogorov-Smirnov tests are modified to remove this difficulty and some properties of this modification are investigated. In particular, a rigorous method for choosing sample size (Theorem 3.2 and corollaries) is presented. Given independent random variables $X_1, \cdots, X_n$ with common distribution function $F$, suppose that we desire to determine whether or not $F$ is in some class $\mathscr{H}_0$. If we are only interested in whether $F$ is close, in some sense, to some distribution function in $\mathscr{H}_0$, we can let $\mathscr{H}^\ast_0 \supset \mathscr{H}_0$, where $\mathscr{H}^\ast_0$ is the class of distribution functions "close" to those in $\mathscr{H}_0$, and test the more reasonable hypothesis that $F \epsilon \mathscr{H}^\ast_0$. In what follows we consider tests based on the uniform metric $d_1$, given by $d_1(F, H) = \sup_x |F(x) - H(x)|$, where $\mathscr{H}_0$ consists of a single distribution function $F_0$.


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Judah Rosenblatt. "Some Modified Kolmogorov-Smirnov Tests of Approximate Hypotheses and their Properties." Ann. Math. Statist. 33 (2) 513 - 524, June, 1962.


Published: June, 1962
First available in Project Euclid: 27 April 2007

zbMATH: 0116.37504
MathSciNet: MR139233
Digital Object Identifier: 10.1214/aoms/1177704577

Rights: Copyright © 1962 Institute of Mathematical Statistics

Vol.33 • No. 2 • June, 1962
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