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On limit distributions of Kolmogorov-Smirnov statistics for a composite hypothesis. (Russian) Zbl 0571.62009

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

MSC:

62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
62F05 Asymptotic properties of parametric tests
60G15 Gaussian processes
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