On the “Poisson boundaries” of the family of weighted Kolmogorov statistics. (English) Zbl 1268.62043

Dasgupta, Anirban (ed.), A Festschrift for Herman Rubin. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-61-7/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 45, 319-331 (2004).
Summary: 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 \(H_1:F(x)\neq F_0(x)\). 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 \(R_n\to_d\sup_{t>0}\mathbb{N}(t)/t\) where \(\mathbb{N}\) is a standard Poisson process on \([0,\infty)\). We call the particular distribution function \(F\) which leads to this limiting Poisson behavior the 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 this 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).
For the entire collection see [Zbl 1066.62002].


62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference