On the "Poisson boundaries" of the family of weighted Kolmogorov statistics
Leah Jager, Jon A. Wellner
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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).
Full-text: Open access
Permanent link to this document: http://projecteuclid.org/euclid.lnms/1196285400
Mathematical Reviews (MathSciNet):
MR2126907
Digital Object Identifier: doi:10.1214/lnms/1196285400
Institute of Mathematical Statistics Lecture Notes - Monograph Series