Some distributional problems connected with order statistics

Two characterizations of distributions symmetric about zero are given. These are based on the distributional properties of the squates of the order statistics from a random sample from these distributions. A result explering the relation between the distribution funcitons of two unordered (not necessarily independent) variables and those of their order statistics is presented. This has some interesting applications.     

First moment approximations for order statistics from the extreme value distribution

Approximate expressions of the first moment of the order statistics of standard extreme value distributions are proposed. We compare different previously given approximations with the exact values. The results show that the approximation given here fits the exact values better than previously given models.     

Outliers and order statistics

Outliers are to be found among the extremes of a data set. Extremes are examples of order statistics. It is thus relevant to ask to what extent the statistical methods (and probabilistic properties) of outliers and of order statistics coincide and depend on each other. Whilst clear overlap is identifiable, aims and procedures are often quite distinct and each topic plays its own important role in the panoply of statistical principles and methodology.     

Hypotheses testing for generalized order statistics with simple order restrictions on model parameters under the alternative

In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score test, and Wald's test for test problems arising from the question of appropriate model selection for ordered data, where simple order restrictions are put on the parameters under the alternative hypothesis. For simple and composite null hypothesis, explicit representations of the corresponding test statistics are obtained along with some properties and their asymptotic distributions. A simulation study is carried out to compare the order restricted tests in terms of their power. In the set-up considered, the adapted tests significantly improve the power of the associated omnibus versions for small sample sizes, especially when testing a composite null hypothesis.     

Confidence intervals for n in the exponential order statistics problem

A simplified proof of the basic properties of the estimators in the Exponential Order Statistics (Jelinski-Moranda) Model is given. The method of constructing confidence intervals from hypothesis tests is applied to find conservative confidence intervals for the unknown parameters in the model.     

Regression via order statistics and their concomitants

We consider a five-dimensional normal distribution and derive the exact joint distribution one variable, linear combinations of order statistics from two other variables, and linear combinations of the corresponding concomitants of these order statistics. We show that this joint distribution is a mixture of trivariate unified skew-normal distributions. This mixture representation enables us to predict one variable based on linear combinations of order statistics from two other variables and linear combinations of the corresponding concomitants. We finally illustrate the usefulness of these results by using a real data.     

Characterization of distributions by conditional expectation of generalized order statistics

In this work, general forms of many well-known continuous probability distributions are characterized by conditional expectation of some functions of generalized order statistics. These results are the generalization of the characterization results based on conditional expectation of the functions of order statistics given by Khan and Abu-Salih (1989).     

On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

Some identities among moments of order statistics

Some new identities among the m oments of order statistics are derived. These are more general in nature and are applicable when moments of Some extreme order statistics do not exist.     

Simulation of Balakrishnan skew-normal order statistics

ABSTRACT

The novel Balakrishnan skew-normal distribution introduced in 2008 has received considerable interest. Here, we derive stochastic representations for simulating order statistics of the novel Balakrishnan skew-normal distribution. The resulting algorithms are more efficient than the ordinary sorting algorithm.     

Domains of attraction of asymptotic distributions of extreme generalized order statistics

Abstract

In extreme value theory for ordinary order statistics, there are many results that characterize the domains of attraction of the three extreme value distributions. In this article, we consider a subclass of generalized order statistics for which also three types of limit distributions occur. We characterize the domains of attraction of these limit distributions by means of necessary and/or sufficient conditions for an underlying distribution function to belong to the respective domain of attraction. Moreover, we compare the domains of attraction of the limit distributions for extreme generalized order statistics with the domains of attraction of the extreme value distributions.     

A note on hazard rates of order statistics

Let F_k:m be the cumulative disribution function of the kth order statistic in a sample of size n from a distribution

F(x) with density function f(x).The primary objective of this paper is to show that F_k+1mis IHR(increasing hazard rate) if F_km(x)is IHH and that F_k-1:n(x)is DHR.(decreasing hazard rate) if F_km(x) is DHR.     

Structure-preserving mappings of uniform order statistics

There are several well-known mappings which transform the first r common order statistics in a sample of size n from a standard uniform distribution to a full vector of dimension r of order statistics in a sample of size r from a uniform distribution. Continuing the results reported in a previous paper by the authors, it is shown that transformations of these types do not lead to order statistics from an i.i.d. sample of random variables, in general, when being applied to order statistics from non-uniform distributions. By accepting the loss of one dimension, a structure-preserving transformation exists for power function distributions.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Characterization of discrete populations through conditional expectations of order statistics

In this paper we give some properties of the expected values of any order statistic when one of its adjacent order statistics is known (order mean function) from a sequence of sizen of independent and identically distributed random variables with discrete distribution. Furthermore, we obtain the explicit expressions of the distribution from these order mean functions, and finally, we show the necessary and sufficient conditions for any real function to be an order mean function. We also add some examples of characterization of discrete distributions from the order mean functions. Partially supported by Consejería de Cultura y Educación (C.A.R.M.), under Grant PIB 95/90.     

Random translation of discrete order statistics

We investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose order statistics are related by a random translation. We also provide some examples using the constructive method that we propose.     

Fisher information in generalized order statistics

A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based representation for samples of k-records which apparently has not been known so far. Moreover, sufficient conditions for the validity of this representation in location and scale family settings are given. The result is illustrated by considering generalized order statistics based on logistic, Laplace, and extreme value distributions.     

Comparison of order statistics in two-sample problem in the sense of Pitman closeness

In this paper, we study the Pitman measure of closeness of order statistics of two independent samples from the same distribution to population quantiles. We then derive various exact expressions of the probability closeness of order statistics from the X and Y samples. Some distribution-free results for the median of the sampling distribution are obtained. Exact and explicit expressions are presented for Uniform(?1, 1) and exponential distributions. Numerical results for illustrative purposes are also provided.     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.