## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 45, 2004, 319-331

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

#### Abstract

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

**Source***A Festschrift for Herman Rubin* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2004)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196285400

**Digital Object Identifier**

doi:10.1214/lnms/1196285400

**Mathematical Reviews number (MathSciNet)**

MR2126907

**Zentralblatt MATH identifier**

1268.62043

**Subjects**

Primary: primary 60G15 60G99: None of the above, but in this section

Secondary: 60E05: Distributions: general theory

**Keywords**

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

**Rights**

Copyright © 2004, Institute of Mathematical Statistics

#### Citation

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. https://projecteuclid.org/euclid.lnms/1196285400