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.     

A note on uniform asymptotic normality of intermediate order statistics

It is proved that under fairly general von Mises-type conditions on the underlying distribution, the intermediate order statistics, properly standardized, converge uniformly over all Borel sets to the standard normal distribution. This closes the gap between central order statistics and extremes, where uniform convergence under mild conditions is well-known.     

Unimodality of uniform generalized order statistics,with applications to mean bounds

We prove that uniform generalized order statistics are unimodal for an arbitrary choice of model parameters. The result is applied to establish optimal lower and upper bounds on the expectations of generalized order statistics based on nonnegative samples in the population mean unit of measurement. The bounds are attained by two-point distributions.     

Characterization through distributional properties of dual generalized order statistics

Distributional properties of two non-adjacent dual generalized order statistics have been used to characterize distributions. Further, one sided contraction and dilation for the dual generalized order statistics are discussed and then the results are deduced for generalized order statistics, order statistics, lower record statistics, upper record statistics and adjacent dual generalized order statistics.     

Rates of uniform convergence of extreme order statistics

Summary Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution. The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.     

SOME COMPARISONS BETWEEN GENERALIZED ORDER STATISTICS

Some stochastic comparisons of generalized order statistics under the right spread order,the location independent riskier order and the total time transform order are investigated in this paper.The underlying distributions and parameters on which generalized order statistics are based are also surveyed to obtain the conditions for increasing the expectations of spacings between the first two generalized order statistics and between the last two generalized order statistics.     

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.     

Some asymptotic expansions of moments of order statistics

Series expansions of moments of order statistics are obtained from expansions of the inverse of the distribution function. They are valid for certain types of distributions with regularly varying tails. We show that the expansions converge quickly when the sample size is moderate to large, and we obtain bounds on the rate of convergence. The special case of the Cauchy distribution is treated in more detail.     

LAN of extreme order statistics

Consider an iid sampleZ ₁,...,Z _n with common distribution functionF on the real line, whose upper tail belongs to a parametric family {F : }. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Z _n–i+1n ) _i=1 ^k of the upperk=k(n) _n order statistics in the sample, if the family {F :} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZ _n–k+1n is the central sequence generating LAN. This implies thatZ _n–k+1n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter can be based on the single order statisticZ _n–k+1n . The rate at whichZ _n–k+1n becomes asymptotically sufficient is however quite poor.     

Some new results on unimodality of generalized order statistics and their spacings

Because of the importance of ordered random variables and, in general, generalized order statistics (GOSs), in many branches of Statistics, a wide interest has been shown in investigating unimodality and strong unimodality of such random variables. Assuming certain restrictions on the model parameters and distributions, some authors have shown unimodality of GOSs. In this article, we shall provide some new results on unimodality of GOSs based on the population distribution function which contain and strengthen several known findings in this regard. A counterexample is also provided for the cases where the results are not valid in general. Unimodality of arbitrary spacings of GOSs based on exponential distributions is also discussed.     

Precise asymptotics for a type of order statistics

This paper obtains some results on the precise asymptotics for the order statistics generated by the random samples of maximum domain of attraction of the Fréchet distribution, which reveal the relations among the boundary function, weight function, convergence rate and limit position in a uniform form.Research supported by National Science Foundation of China (NO. 10271087).     

Lorenz ordering of power-function order statistics

Based on Lorenz comparisons of two random variables from the four-parameter generalized beta distribution of the first kind, Lorenz-order relationships among order statistics from different power-function samples with varying sizes are obtained.     

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     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

Ordering distributions by scaled order statistics

Motivated by applications in reliability theory, we define a preordering (X ₁, ...,X _n) (Y ₁ ...,Y _n) of nonnegative random vectors by requiring thek-th order statistic ofa ₁ X ₁,..., a _n X _n to be stochastically smaller than thek-th order statistic ofa ₁ Y ₁, ...,a _n Y _n for all choices ofa _i >0,i=1, 2, ...,n. We identify a class of functionsM _{k, n} such that if and only ifE(X)E(Y) for allM _k,n. Some preservation results related to the ordering are obtained. Some applications of the results in reliability theory are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205.     

Approximation to the expectation of a function of order statistics and its applications

ThisprojectissupportedbytheNationalNaturalScienceFoundationofChinaandDoctoralProgramFoundationofHigherEducation.1.IntroductionLetUI,U2,'bei.i.d.randomvariableswithuniformd.f.ontheinterval(0,l),andforeveryn31,writeUt,,15'5Un,.fortheorderstatisticsofUI,'tUn.SupposethatXI1X2,'arei.i.d.observationsfromanondegenerated.f.F,anddenotebyX.,l5'5X.,.theorderstatisticsofXI,'IX,,'Withoutlossofgenerality,wewillassume0相似文献   

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.     

Maximum variance of order statistics

Yang (1982,Bull. Inst. Math. Acad. Sinica,10(2), 197–204) proved that the variance of the sample median cannot exceed the population variance. In this paper, the upper bound for the variance of order statistics is derived, and it is shown that this is attained by Bernoulli variates only. The proof is based on Hoeffding's identity for the covariance.     

Rank order statistics for time series models

Some invariance principles are obtained for the one-sample rank order statistics of a -mixing or strong mixing type time series. The estimation of the center of symmetry of the time series and tests for serial dependence are considered as applications.     

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.