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

Nevzorov, V. B.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 130, 137-146 (1983).

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

Keywords:

order statistics; not identically distributed variables; independent variables; quantile; normal distribution; differentiable density