Institute of Mathematical Statistics Lecture Notes - Monograph Series

On the "Poisson boundaries" of the family of weighted Kolmogorov statistics

Leah Jager and Jon A. Wellner

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Berk and Jones (1979) introduced a goodness of fit test statistic $R_n$ which is the supremum of pointwise likelihood ratio tests for testing $H_0 : F(x) = F_0 (x)$ versus $H_1 : F (x) \not= F_0 (x)$. They showed that their statistic does not always converge almost surely to a constant under alternatives $F$, and, in fact that there exists an alternative distribution function $F$ such $R_n \rightarrow_d \sup_{t>0} \NN(t)/t$ where $\NN$ is a standard Poisson process on $[0,\infty)$. We call the particular distribution function $F$ which leads to this limiting Poisson behavior the {\sl Poisson boundary distribution function for} $R_n$. We investigate Poisson boundaries for weighted Kolmogorov statistics $D_n (\psi)$ for various weight functions $\psi$ and comment briefly on the history of results concerning Bahadur efficiency of these statistics. One result of note is that the logarithmically weighted Kolmogorov statistic of Groeneboom and Shorack (1981) has the same Poisson boundary as the statistic of Berk and Jones (1979).

Chapter information

Anirban DasGupta, ed., A Festschrift for Herman Rubin (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2004), 319-331

First available in Project Euclid: 28 November 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: primary 60G15 60G99: None of the above, but in this section
Secondary: 60E05: Distributions: general theory

Bahadur efficiency Berk-Jones statistic consistency fixed alternatives goodness of fit Kolmogorov statistic Poisson process power weighted Kolmogorov statistic

Copyright © 2004, Institute of Mathematical Statistics


Jager, Leah; Wellner, Jon A. On the "Poisson boundaries" of the family of weighted Kolmogorov statistics. A Festschrift for Herman Rubin, 319--331, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2004. doi:10.1214/lnms/1196285400.

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