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The convergence rate of order statistics to the normal law for not identically distributed variables. (Russian. English summary) Zbl 0532.62009

Let \(X_ 1,X_ 2,...,X_ n\) be independent random variables, and let \(X_{r:n}\) be the r-th order statistic of the \(X_ j\). The author gives a uniform estimate for the deviation of the distribution of \((X_{r:n}- A_ n^{(r)})/B_ n^{(r)}\) from the normal distribution under the following conditions: \(X_ j\) has a differentiable density function \(f_ j(x)\) and f’\({}_ j(x)\) is bounded by a number M, not depending on j. The normalizing constants \(A_ n^{(r)}\) and \(B_ n^{(r)}>0\) are explicitly given. The estimate is meaningful only if both r and n-r tend to infinity with n. The author indicates that extension to the multivariate distribution of \((X_{r_ j:n}-A_{n,j})/B_{n,j},\) 1\(\leq j\leq k\), can be given on the line of the univariate case.
Reviewer: J.Galambos

MSC:

62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
60F05 Central limit and other weak theorems
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