Bounds on dispersion of order statistics based on dependent symmetrically distributed random variables

We consider a fixed number of arbitrarily dependent random variables with a common symmetric marginal distribution. For each order statistic based on the variables, we determine a common optimal bound, dependent in a simple way on the sample size and number of order statistics, for various measures of dispersion of the order statistics, expressed in terms of the same dispersion measure of the single original variable. The dispersion measures are connected with the notion of M-functional of a random variable location with respect to a symmetric and convex loss function. The measure is defined as the expected loss paid for the discrepancy between the M-functional and the variable. The most popular examples are the median absolute deviation and variance.     

Multivariate testing and model-checking for generalized order statistics with applications

The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Wald's test, and Rao's score test are derived and turn out to have simple representations. The asymptotic distribution of the corresponding test statistics under the null hypothesis is stated, and, in case of a simple null hypothesis, asymptotic optimality of the LR test is addressed. Applications of the tests are presented; in particular, we discuss their use in reliability, and to decide whether a Poisson process is homogeneous. Finally, a power study is performed to measure and compare the quality of the tests for both, simple and composite null hypotheses.     

Sharp bounds on l-estimates and their expectations for dependent samples

Sharp bounds on the expectation of each L-estimate are determined for samples of identically distributed random variables with a given expectation and a central absolute moment of order p > 1, The distributions achieving these bounds are constructed. Analogous results are derived in the cases of symmetric, noraiegative and bounded random variables, Accurate bounds for L-estimates of arbitrary finite samples are concluded.     

Application of Concomitants of Order Statistics of Independent Non Identically Distributed Random Variables in Estimation

In this article, we obtain expressions for the pdf of a single concomitant of order statistic and the joint pdf of a pair of concomitants of order statistics of independent non identically distributed random variables. Using these expressions, we find the means, variances and covariances of order statistics arising from independent non identically distributed bivariate Pareto distributions. A method of estimation of a common parameter involved in several bivariate Pareto distributions using concomitants of order statistics is also discussed.     

Bounds on expected generalized order statistics

We establish the upper nonpositive and all the lower bounds on the expectations of generalized order statistics based on a given distribution function with the finite mean and central absolute moment of a fixed order. We also describe the distributions for which the bounds are attained. The methods of deriving the lower nonpositive (upper nonnegative) and lower nonnegative (upper nonpositive) bounds are totally different. The first one, the greatest convex minorant method is the combination of the Moriguti and well-known Hölder inequalities and the latter one is based on the maximization of some norm on the properly chosen convex set. The paper completes the results of Cramer et al. [Evaluations of expected generalized order statistics in various scale units. Appl Math. 2002;29:285–295].     

Sequential order statistics from dependent random variables

Abstract

This paper provides an extension for “sequential order statistics” (SOS) introduced by Kamps. It is called “developed sequential order statistics” (DSOS) and is useful for describing lifetimes of engineering systems when component lifetimes are dependent. Explicit expressions for the joint density function, the marginal distributions and the means of DSOS are derived. Under the well known “conditional proportional hazard rate” (CPHR) model and the Gumbel families of copulas for dependency among component lifetimes, some findings are reported. For example, it is proved that the joint density functions of DSOS and SOS have the same structure. Various illustrative examples are also given.     

Stochastic comparisons of order statistics,with applications in relibility

This paper reviews recent developments in the stochastic comparison of order statistics. The results discussed are basically: (l) Stochastic comparisons of linear combinations of order statistics from distributions F and G where G^?1 F is convex or starshaped. (2) Stochastic comparisons of individual order statistics and of vectors of order statistics from underlying heterogeneous distributions by the use of majorization and Schur function theory. (3) Stochastic comparison of random processes. Applications to reliability problems are presented illustrating the use and value of the theoretical results described     

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.     

Prediction of order statistics and record values from two independent sequences

In this paper, we discuss the problem of predicting future order statistics based on observed record values and similarly, the prediction of future records based on observed order statistics. The coverage probabilities of these intervals are exact and are free of the parent distribution F. Finally, two data sets are used to illustrate the proposed procedures.     

Asymptotic properties of numbers of observations in random regions determined by central order statistics

In this paper, we extend the concept of near order statistic observation by considering observations that fall into a random region determined by a given order statistic and a Borel set. We study asymptotic properties of numbers of such observations as the sample size tends to infinity and the order statistic is a central one. We show that then proportions of these numbers converge in probability to some population probabilities. We also prove that these numbers can be centered and normalized to yield normal limit law. First, we derive results for one order statistic; next we give extensions to the multivariate case of two or more order statistics.     

The asymptotic distribution of numbers of observations near order statistics

In this paper, we discuss the limiting behavior of numbers of observations near an order statistic. We then derive an expression for the joint distribution of the numbers of observations that fall into the open right a-vicinity and left b-vicinity of k  th and (n-r)$(n-r)$th order statistics, respectively, from a sample of size n and establish the result that they are asymptotically independent under suitable conditions.     

Limit theory of bivariate dual generalized order statistics with random index

The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted.     

Asymptotic behaviour of proportions of observations in random regions determined by central order statistics from stationary processes

In this paper, we show that proportions of observations that fall into a random region determined by a given Borel set and a central order statistic converge almost surely, provided that the corresponding population quantile is unique. We also describe three types of possible asymptotic behaviour of these proportions in the case of non-unique population quantile. As an application of our findings we establish limiting properties of numbers of ties with a central order statistics in a discrete sample. Our results are derived not only for independent and identically distributed observations but more generally for strictly stationary and ergodic sequences of random variables.     

The strong consistency of M-estimates in linear models with extended negatively dependent errors

In this paper, we first establish the strong convergence for weighted sums of extended negatively dependent (END) random variables. Based on the strong convergence and Bernstein inequality, we obtain the strong consistency of M-estimates of the regression parameters in a linear model for END random errors under some mild moment conditions. The results generalize and improve the ones obtained in the literature to the case of END random errors.     

Higher order moments of order statistics from INID symmetric random variables

In this paper we present analogues of Balakrishnan's (1989) relations that relate the triple and quadruple moments of order statistics from independent and nonidentically distributed (I.NI.D.) random variables from a symmetric distribution to those of the folded distribution. We then apply these results, along with the corresponding recurrence relations for the exponential distribution derived recently by Childs (2003), to study the robustness of the Winsorized variance.     

Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors

In this paper, we provide some exponential inequalities for extended negatively dependent (END) random variables. By using these exponential inequalities and the truncated method, we investigate the complete consistency for the estimator of nonparametric regression model based on END errors. As an application, the complete consistency for the nearest neighbour estimator is obtained.     

Table of percentage points of ratios of linear combinations of order statistics of samples from exponential distributions

A form of the distribution function of ratios of linear combinations of order statistics of samples from an exponential distribution is given. From the distribution, tables of percentage points of the statistic for α = .05, .95, and n = 3(1)50, and for censoring up to five observations are presented. Use of the tables is made to find critical values of the most powerful scale and location invariant test of exponentiality against uniformity, and also to find critical values for a test of outliers in an exponential population.