On the “Poisson boundaries” of the family of weighted Kolmogorov statistics

Berk and Jones (1979) introduced a goodness of fit test statistic Rn which is the supremum of pointwise likelihood ratio tests for testing H0 : F (x) = F0(x) versus H1 : F (x) = F0(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 Rn →d supt>0 N(t)/t where N is a standard Poisson process on [0,∞). We call the particular distribution function F which leads to this limiting Poisson behavior the Poisson boundary distribution function for Rn. We investigate Poisson boundaries for weighted Kolmogorov statistics Dn(ψ) for various weight functions ψ 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).