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On the asymptotic distribution of weighted uniform empirical and quantile processes in the middle and on the tails. (English) Zbl 0584.62025

Let \((\alpha_ n)\) and \((u_ n)\), respectively, denote the uniform empirical and the uniform quantile processes. In this paper, the authors obtain, for \(\nu =0\) and \(0<\nu \leq 1/2\), the asymptotic distributions of \[ \sup | \alpha_ n(s)| /(s(1-s))^{1/2-\nu}\quad and\quad \sup | u_ n(s)| /(s(1-s))^{1/2-\nu}, \] properly normalized, when the supremum is taken over subintervals of [0,1], near the tails and in the middle, depending on n.
The technique is to obtain similar asymptotic distributions for sequences of Brownian bridges and to claim the same for the above processes through ’a sort of weak invariance theorem’. Further, the asymptotic independence of these processes is discussed.
When \(-1/2\leq \nu <0\), the asymptotic distribution near the tails can be deduced from a result of the second author, ibid. 15, 99-109 (1983; Zbl 0504.62021), and that for the middle is obtained in the second author, The asymptotic distribution of generalized Renyi statistics. Acta. Sci. Math. 48, 329-338 (1985). When \(\nu =1/2\), D. Jaeschke [Ann. Stat. 7, 108-115 (1979; Zbl 0398.62013)] has obtained similar results for intervals in the middle.
Reviewer: R.Vasudeva

MSC:

62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
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