Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.     

Role of concomitants of order statistics in determining parent bivariate distributions

In this paper, we establish the role of concomitants of order statistics in the unique identification of the parent bivariate distribution. From the results developed, we have illustrated by examples the process of determination of the parent bivariate distribution using a marginal pdf and the pdf of either of the concomitant of largest or smallest order statistic on the other variable. An application of the results derived in modeling of a bivariate distribution for data sets drawn from a population as well is discussed.     

Correlation analysis of ordered symmetrically dependent observations and their concomitants of order statistics

Given two jointly observed random vectors Y and Z of the same dimension, let Y be a reordered version of Y and Z the resulting vector of concomitants of order statistics. When X is a covariate of interest, also jointly observed with Y, the authors obtain the joint covariance structure of (X, y, Z) and the related correlation parameters explicitly, under the assumption that the vector (X, Y, Z) is normal and that its joint covariance structure is permutation symmetric. They also discuss extensions to elliptically contoured distributions.     

On an application of concomitants of order statistics

ABSTRACT

Let (X_i, Y_i), i = 1, …, n be a pair where the first coordinate X_i represents the lifetime of a component, and the second coordinate Y_i denotes the utility of the component during its lifetime. Then the random variable Y_{[r: n]} which is known to be the concomitant of the rth order statistic defines the utility of the component which has the rth smallest lifetime. In this paper, we present a dynamic analysis for an n component system under the above-mentioned concomitant setup.     

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.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Expected sample moments of concomitants of selected order statistics

In this paper, the task of determining expected values of sample moments, where the sample members have been selected based on noisy information, is considered. This task is a recurring problem in the theory of evolution strategies. Exact expressions for expected values of sums of products of concomitants of selected order statistics are derived. Then, using Edgeworth and Cornish-Fisher approximations, explicit results that depend on coefficients that can be determined numerically are obtained. While the results are exact only for normal populations, it is shown experimentally that including skewness and kurtosis in the calculations can yield greatly improved results for other 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.     

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.     

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.     

On an Application of Concomitants of Order Statistics in Characterizing a Family of Bivariate Distributions

In this article, we consider a family of bivariate distributions which includes the well-known Morgenstern family of bivariate distributions as its subclass. We identify some properties of concomitants of order statistics which characterize this generalized class of distributions. An application of the characterization result in modeling a bivariate distribution to a data is also explained.     

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution. The distributions of the minimum and maximum are obtained as special cases.     

Estimation of parameters of bivariate normal distribution using concomitants of record values

In this paper, we discuss the concomitants of record values arising from the well-known bivariate normal distribution BVND(μ₁, μ₂,σ₁,σ₂, ρ). We have obtained the best linear unbiased estimators of μ₂ and σ₂ when ρ is known and derived some unbiased linear estimators of ρ when μ₂ and σ₂ are known, based on the concomitants of first n record values. The variances of these estimators have been obtained.     

On generalized order statistics from linear exponential distribution and its characterization

In this paper, recurrence relations for single and product moments of generalized order statistics (gOSs) from linear exponential distribution (LE) are derived and characterizations of this distribution based on the conditional moments of the gOSs are given.     

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.     

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.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

Simultaneous Pitman closeness of progressively type-II right-censored order statistics to population quantiles

In this work, we extend prior results concerning the simultaneous Pitman closeness of order statistics (OS) to population quantiles. By considering progressively type-II right-censored samples, we derive expressions for the simultaneous closeness probabilities of the progressively censored OS to population quantiles. Explicit expressions are deduced for the cases when the underlying distribution has bounded and unbounded supports. Illustrations are provided for the cases of exponential, uniform and normal distributions for various progressive type-II right-censoring schemes and different quantiles. Finally, an extension to the case of generalized OS is outlined.