On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.     

On the limiting joint distribution of the extreme order statistics

The connection between extreme values and record-low values is exploited to derive simply the limiting joint distribution of the r largest order statistics. The use of this distribution in the modelling of corrosion phenomena is considered, and the extrapolation of maxima in space and time is described in this context. There has been recent emphasis on movement away from classical extreme value theory to more efficient estimation procedures. This shift is continued with the illustration of the extra precision of predicted maxima obtained from a model based on extreme order statistics over the classical extreme value approach.     

On the limit behaviour of the joint distribution function of order statistics

Fork ₀ fixed we consider the joint distribution functionF _n ^k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF _n ^k (x ₁,,x _n-k) in terms of the limit behaviour ofn(1-F(x _n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution functionF.     

A note on the moments of order statistics from doubly truncated exponential distribution

Summary In a recent paper [2], the author has obtained some recurrence relations between the moments of order statitics from the exponential and right truncated exponential distributions. In this paper, similar relations are derived for a doubly truncated exponential distribution. It is shown that one can obtain all the moments by using these recurrence relations.     

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.     

The exact and approximate distributions of linear combinations of selected order statistics from a uniform distribution

Summary The exact probability density function is given for linear combinations ofk=k(n) order statistics selected from whole order statistics based on random sample of sizen drawn from a uniform distribution. Normal approximation to the linear combinations is made with the aid of Berry-Esseen's theorem. Necessary and sufficient conditions of the asymptotic normality for the statistic are obtained, too. An exact distribution and its normal approximation of linear combination of mutually independent gamma variables with integer valued parameters are also given as associated consequences. The Institute of Statistical Mathematics     

On asymptotic distributions of exceedance statistics

In this study, two independent samples X₁,X₂,…,X_n and Y₁,Y₂,…,Y_m with respective distribution functions F and Q are considered. The joint asymptotic distributions of exceedance statistics defined as the number of Y observations falling into a random interval of order statistics constructed from the X sample is investigated. The results can be used in the context of a two-sample problem.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

On a problem of Arnold on uniform distribution

In this note we consider some quantitative versions of conjectures made by Arnold related to Galois dynamics in finite fields. We refine some results by Shparlinski using exponential sum results.     

On the minimum of several random variables

For a given sequence of real numbers , we denote the th smallest one by . Let be a class of random variables satisfying certain distribution conditions (the class contains Gaussian random variables). We show that there exist two absolute positive constants and such that for every sequence of real numbers and every , one has

-

where are independent random variables from the class . Moreover, if , then the left-hand side estimate does not require independence of the 's. We provide similar estimates for the moments of as well.

     

Relationships for moments of order statistics from the right-truncated generalized half logistic distribution

In this paper, we establish several recurrence relations satisfied by the single and the product moments for order statistics from the right-truncated generalized half logistic distribution. These relationships may be used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes and for any choice of the truncation parameter P. These generalize the corresponding results for the generalized half logistic distribution derived recently by Balakrishnan and Sandhu (1995, J. Statist. Comput. Simulation, 52, 385–398).Earlier went by the name R. A. Sandhu.     

On the exact distribution and mean value function of a geometric process with exponential interarrival times

The geometric process is considered when the distribution of the first interarrival time is assumed to be exponential. An analytical expression for the one dimensional probability distribution of this process is obtained as a solution to a system of recursive differential equations. A power series expansion is derived for the geometric renewal function by using an integral equation and evaluated in a computational perspective. Further, an extension is provided for the power series expansion of the geometric renewal function in the case of the Weibull distribution.     

On the jackknife statistics for eigenvalues and eigenvectors of a correlation matrix

This paper deals with some problems of eigenvalues and eigenvectors of a sample correlation matrix and derives the limiting distributions of their jackknife statistics with some numerical examples.     

On the value distribution of f a(f')~n

Let f be a transcendental meromorphic function,a a nonzero finite complex number,and n 2 a positive integer.Then f a(f')n assumes every complex value infinitely often.This answers a question of Ye for n = 2.A related normality criterion is also given.     

均匀分布位置参数的估计量的渐近分布及应用

对区间长度为定值均匀分布位置参数的点估计量进行了研究,得到位置参数点估计量的渐近分布.讨论了渐近分布的相关性质.给出了位置参数的区间估计及其假设检验方法.     

On a logarithmic integral and the moments of order statistics from the Weibull-geometric and half-logistic families of distributions

We provide an explicit analytical solution for a logarithmic integral in terms of the Lerch transcendent function together with the generalized Stirling numbers of the first kind. For some special cases of interest in statistical applications, the explicit solution can be expressed in terms of the polylogarithm function together with the aforementioned Stirling numbers. As a consequence, we obtain explicit expressions for the moments of order statistics from the half-logistic distribution, the Weibull-geometric distribution and the long-term Weibull-geometric distribution, which include as particular cases the extended exponential-geometric distribution and the long-term extended exponential-geometric distribution, among others. These analytical expressions are useful for computational purposes.     

A Laplace transform method for order statistics from nonidentical random variables and its application in Phase-type distribution

In this paper, we derive a method for obtaining the Laplace transform of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.’s). A survey of the most important properties, applications and the o.s. of a Phase-type (PH) distribution are also presented. Two illustrative examples are provided.     

Computing the moments of order statistics from nonidentically distributed phase-type random variables

In this paper, we derive a recurrence relation for the single moments of order statistics (o.s.) arising from n independent nonidentically distributed phase-type (PH) random variables (r.v.’s). This recurrence relation will enable one to compute all single moments of all o.s. in a simple recursive manner.