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.     

Inequalities for the partial sums of elliptical order statistics related to genetic selection

Let X₁,…,X_n be exchangeable normal variables with a common correlation p, and let X₍₁₎ > … > X_(n) denote their order statistics. The random variable σⁿ_i=_n—_k+1x_i, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.     

On characterizations based on record values and order statistics

Consider an infinite sequence of independent random variables having common continuous c.d.f. F. For 1 ⩽ i ⩽ n, let X_i:n denote the ith order statistic of the first n random variables, and let {X(n), n ⩾ 1} be the sequence of upper record values. We examine the similarities and differences between the dependence structures of the X_i:n's and the X(n)'s, with an emphasis on the latter. We present an interesting situation involving a characterization of F using the moment sequence of records. We obtain characterizations based on the properties of certain regression functions associated with order statistics, record values, and the original observations. We discuss the resemblance between some known and some new characterizations based on order statistics, record values and those based on the properties of truncated F.     

Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

In this paper we consider a sequence of independent continuous symmetric random variables X₁, X₂, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages S_n = R⁽ⁿ⁾₁X₁ + ??? + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 ? 1/α}/Γ^1/α(α + 1).     

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.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.     

On Dependence Properties of Random Minima and Maxima

Let X _(n) and X ₍₁₎ be the largest and smallest order statistics, respectively, of a random sample of fixed size n. Quite generally, X ₍₁₎ and X _(n) are approximately independent for n sufficiently large. In this article, we study the dependence properties of random extremes in terms of their copula, when the sample size has a left-truncated binomial distribution and show that they tend to be more dependent in this case. We also give closed-form formulas for the measures of association Kendall's τ and Spearman's ρ to measure the amount of dependence between two extremes.     

Information Properties for Concomitants of Order Statistics in Farlie–Gumbel–Morgenstern (FGM) Family

Let (X _i , Y _i ), i = 1, 2,…, n be independent and identically distributed random variables from some continuous bivariate distribution. If X _(r) denotes the rth-order statistic, then the Y's associated with X _(r) denoted by Y _[r] is called the concomitant of the rth-order statistic. In this article, we derive an analytical expression of Shannon entropy for concomitants of order statistics in FGM family. Applying this expression for some well-known distributions of this family, we obtain the exact form of Shannon entropy, some of the information properties, and entropy bounds for concomitants of order statistics. Some comparisons are also made between the entropy of order statistics X _(r) and the entropy of its concomitants Y _[r]. In this family, we show that the mutual information between X _(r) and Y _[r], and Kullback–Leibler distance among the concomitants of order statistics are all distribution-free. Also, we compare the Pearson correlation coefficient between X _(r) and Y _[r] with the mutual information of (X _(r), Y _[r]) for the copula model of FGM family.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

A characterization of the geometric distribution

We show that for a simple random sample from a discrete distribution on the positive integers, the regression ofX _(2∶n) onX _(1∶n) is linear with unit slope if and only if the distribution is geometric.     

On some joint distributions in fluctuation theory

For non-negative integral valued interchangeable random variables v₁, v₂,…,v_n, Takács (1967, 70) has derived the distributions of the statistics ?_n' ?¹_n' ?^(c)_n and ?^(-c)_n concerning the partial sums N_r = v₁ + v₂ + ··· + v_rr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α^(c)_n, δ^(c)_n, Z_n), (β^(c)_n, Z_n) and (β^(-c)_n, Z_n) concerning the partial sums N_r = ε₁ + ··· + ε_rr = 1,2,…,n, of geometric random variables ε₁, ε₂,…,ε_n.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

ON CHARACTERIZING DISTRIBUTIONS VIA LINEARITY OF REGRESSION FOR ORDER STATISTICS

Let X₁X_n be a random sample from an absolutely continuous distribution with the corresponding order statistics X_1:n≤X_2:n≤X_n:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of X_k+2:n with respect to X_k:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.     

A formula for P(bi≤Ri≤ai, 1≤i≤m) under a general class of alternatives

Let X_i≤?≤X_m and Y_i≤?≤Y_n be two sets of independent order statistics from continous distributions with distribution functions F and G respectively. Let R_i denote the rank of X_i in the combined order sample. Steck (1980) has found an expression for P(b_i≤R_i≤a_i, all i) when F = h(G), h being the incomplete beta function with parameters (α,β?α+1). An alternative expression for the same probability is obtained which is computationally a substantial improvement on Steck's result.     

Concomitants in a Sequence of Independent Nonidentically Distributed Random Vectors

ABSTRACT

Concomitants of order statistics are considered for the situation in which the random vectors (X ₁, Y ₁), (X ₂, Y ₂),…, (X _n , Y _n ) are independent but otherwise arbitrarily distributed. The joint and marginal distributions of the concomitants of order statistics and stochastic comparisons among the concomitants of order statistics are studied in this situation.     

A unified approach to characterization problems using conditional expectations

In recent years characterization problems have become of increasing interest. It is well known that mean residual life e(x) = E(X - x|X ⩾x) and right-censored mean function m_R(x) = E(X | X ⩽ x), uniquely determine the distribution function F(x) = P(X ⩽ x). In this paper, we study characterizations problems for general distributions, using the doubly censored mean function m(x, y) = E(X | x⩽X⩽y). We show that m(x, y) characterizes F(x), obtaining the explicit expression of F(x) from m(x, y). Moreover, we give properties that any function must verifies to be a doubly censored mean function and we obtain stability theorems for these characterizations.     

COMPARISON OF THREE NON-PARAMETRIC TESTS FOR BIVARIATE INTERCHANGEABILITY1

Given a random sample(X₁, Y₁), …,(X_n, Y_n) from a bivariate (BV) absolutely continuous c.d.f. H (x, y), we consider rank tests for the null hypothesis of interchangeability H₀: H(x, y). Three linear rank test statistics, Wilcoxon (W_N), sum of squared ranks (SSR_N) and Savage (S_N), are described in Section 1. In Section 2, asymptotic relative efficiency (ARE) comparisons of the three types of tests are made for Morgenstern (Plackett, 1965) and Moran (1969)BV alternatives with marginal distributions satisfying G(x) = F(x/θ) for some θ≠ 1. Both gamma and lognormal marginal distributions are used.