On limit distributions for intermediate order statistics under power normalization

In this paper the work of Chibisov (1964) for the intermediate order statistics, under linear normalization, is extended to the power normalization. The possible limits and the corresponding domains of attraction are derived.     

On the restricted convergence of generalized extreme order statistics

Generalized order statistics (gos) introduced by Kamps [8] as a unified approach to several models of order random variables (rv’s), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos, included oos and sos, the possible limit distribution functions (df’s) of the maximum gos are obtained in Nasri-Roudsari [10]. In this paper, for this subclass, as the df of the suitably normalized extreme gos converges on an interval [c, d] to one of possible limit df’s of the extreme gos, the continuation of this (weak) convergence on the whole real line to this limit df is proved.     

On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

On the rate of convergence to normality for generalized linear rank statistics

Summary A generalized linear rank statistic is introduced to include, as special cases, both signed as well as unsigned linear rank statistics. For this statistic, the rate of convergence to asymptotic normality is investigated. It is shown that this rate is of orderO(N ^−1/2 logN) if the score generating function ϕ is twice differentiable, and it is of orderO(N ^−1/2) if the second derivative of ϕ satisfies Lipschitz's condition of order ≧1/2. The results obtained extend as well as generalize most of the earlier results obtained in this direction.     

Higher order asymptotic behaviour of partial maxima of random sample from generalized Maxwell distribution under power normalization

In this article, the higher order asymptotic expansions of cumulative distribution function and probability density function of extremes for generalized Maxwell distribution are established under nonlinear normalization. As corollaries, the convergence rates of the distribution and density of maximum are obtained under nonlinear normalization.     

A note on the convergence of trivariate extreme order statistics and extension

Necessary and sufficient conditions, under which there exists (at least) a sequence of vectors of real numbers for which the distribution function (d.f.) of any vector of extreme order statistics converges to a nondegenerate limit, are derived. The interesting thing is that these conditions solely depend on the univariate marginals. Moreover, the limit splits into the product of the limit univariate marginals if all the bivariate marginals of the trivariate d.f., from which the sample is drawn, is of negative quadrant dependent random variables (r.v.'s). Finally, all these results are stated for the multivariate extremes with arbitrary dimensions.     

The uniform convergence and precise asymptotics of generalized stochastic order statistics

Let X(i,n,m,k), i=1,…,n, be generalized order statistics based on F. For fixed r∈N, and a suitable counting process N(t), t>0, we mainly discuss the precise asymptotic of the generalized stochastic order statistics X(N(n)−r+1,N(n),m,k). It not only makes the results of Yan, Wang and Cheng [J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci. 26 (2) (2006) 237-244] as the special case of our result, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables.     

On the asymptotic simultaneous distribution of randomly indexed order statistics

The asymptotic simultaneous distribution of normalized central order statistics for the random size of sample is studied. This study develops works [1, 2], in which Student’s distribution acts as the limit distribution for a class of statistics. Results from the study are applied to construct statistical insights on the shift/scale parameter ratio for two-parametric families of distributions.     

Estimates of the rate of convergence of sums of order statistics to the normal law. II

Some results are obtained on the rate of convergence of trimmed means to the normal Law. The work continues the investigations begun in the paper [3].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 165–174, 1976.     

On a characterization of the uniform distribution by generalized order statistics

In this paper a new variant of the Choquet-Deny theorem is obtained and used to prove a characterization of the uniform distribution based on spacings of generalized order statistics. This result extends two recent characterizations of the uniform distribution.     

Rates of convergence of extreme for general error distribution under power normalization

By using the theory of p-max stable laws, we study the rates of convergence of extremes for general error distribution under power normalization. We derived the exact uniform convergence rate of the distribution of maximum to its extreme value limit.     

On the rate of convergence of the distributions of certain statistics to the Laplace distribution

An estimation of the rate of convergence of the distributions of asymptotically normal statistics based on a random sample of random size to the Laplace distribution is obtained. It is assumed that the random size of a sample does not depend on the members of the sample and has a number of special properties.     

Uniform convergence of sums of order statistics to stable laws

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