Tyurin, Yu. N. On limit distributions of Kolmogorov-Smirnov statistics for a composite hypothesis. (Russian) Zbl 0571.62009 Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 6, 1314-1343 (1984). Let \(F_ n(x)\) be the empirical distribution function based on n observations from a population with distribution function F(x,\(\theta)\), where the form of F(x,\(\theta)\) is known, but the (r-dimensional) parameter \(\theta\) is unknown. Define \(D^+_ n=\sup_ x(F_ n(x)-F(x,{\hat \theta}))\), where \({\hat \theta}\) is a (specific) estimator of \(\theta\). It has been known that the asymptotic distribution of \(\sqrt{n}D^+_ n\) coincides with the distribution of \(\sup_{0\leq t\leq 1}\omega (t)\), where \(\omega\) (t) is a (special conditional) Wiener process. This latter distribution, however, has not been known. In the present paper, an effective expansion of the distribution of sup \(\omega\) (t) is given. Reviewer: J.Galambos Cited in 1 ReviewCited in 2 Documents MSC: 62E20 Asymptotic distribution theory in statistics 60F05 Central limit and other weak theorems 62F05 Asymptotic properties of parametric tests 60G15 Gaussian processes Keywords:limit distributions; Kolmogorov-Smirnov statistics; composite hypothesis; supremum of Wiener process; empirical distribution; expansion PDFBibTeX XMLCite \textit{Yu. N. Tyurin}, Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 6, 1314--1343 (1984; Zbl 0571.62009)