Bivariate Order Statistics Research Articles

An exceedance model based on bivariate order statistics

In hydrologic risk analysis, the use of exceedance statistics are very important. In this sense, we construct a random threshold model based on bivariate order statistics. The exact distribution of exceedance statistics is calculated under some well-known copulas such as independent and Farlie-Gumbel-Morgenstern (FGM) copulas. Furthermore, numerical results are provided for expected value and variance of exceedance statistics under independent and Farlie-Gumbel-Morgenstern copulas. The application of the model in hydrology is also discussed.

Reliability analysis of a multi-state system with identical units having two dependent components

The performance evaluation of a system having n identical units, each of which consists of two components has been successfully discussed in binary-state reliability analysis. In this paper, we study the performance evaluation of a multi-state system based on bivariate order statistics. The multi-state system consists of n independent and identical units, each having two components. The components of each unit are assumed to be s-dependent. However, the units work s-independently with each other. The system and each component of each unit having three performance levels “0 (failure), 1 (partially working) and 2 (completely working)” are considered. The degradation of the components follows Markov Process and also Farlie-Gumbel-Morgenstern distribution is used to model the s-dependence of the components. The reliability analysis of a multi-state k-out-of- n system are evaluated under the assumptions. Some dynamic performance measures for the system such as the mean residual and mean past lifetime functions based on bivariate order statistics are also evaluated. The performance of the system is especially examined for different values of s-dependence parameter, the degradation rates and different number of units for the system. The results are supported with some numerical examples and graphical representations.

Limit theorems for univariate and bivariate order statistics with variable ranks

In this work, we investigate the asymptotic behaviour of the intermediate and central order statistics of a bivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm is used, we reveal the interrelation between the R-ordering principle and Marginal-Ordering Principle (M-ordering). The asymptotic behaviour of the intermediate and central bivariate order statistics based on sup-norms is completely determined. Besides, we highlight the role of the normalizing constants in characterizing the limit types of order statistics and new results concerning the intermediate order statistics are given.

On the joint distribution of order statistics from independent non-identical bivariate distributions

In this note, the exact joint probability density function (jpdf) of bivariate order statistics from independent non-identical bivariate distributions is obtained. Furthermore, this result is applied to derive the joint distribution of a new sample rank obtained from the rth order statistics of the first component and the sth order statistics of the second component.

On the convergence of bivariate order statistics: Almost sure convergence and convergence rate

We study the convergence of bivariate order statistics, and get the almost sure convergence and convergence rate, generalizing one of main results in Huang et al. (2013). To be more precise, we prove the almost sure convergence of bivariate order statistics without the positive quadrant dependent ( PQD ) condition mentioned in Huang et al. (2013), and extend this result to a more general case. We also give the bound on the Kolmogorov distance between the distributions of bivariate order statistics and its limit. Our results provide an approximate algorithm for sampling from complicated structures, verified by some examples in the last section of this paper.

A Study on the Multi-state (r, s)-out-of- n Systems with Dependent Components

In the study of technical systems in reliability engineering, multi-state systems play a useful role. A multi-state system is a system consisting n components that system and its components may have several level performance. In the present paper, we introduce the multi-state (r, s)-out-of-n system consisting n elements having the property that each element consists two dependent components and each component of the elements and the system can be in one of m+1 possible states: 0, 1, 2,..., m. We investigate an efficient method to compute the exact reliability by using the distribution of bivariate order statistics. Depending on the number of active components of the multi-state (r, s)-out-of-n systems at time t, the mean residual lifetime function of the system is studied. Also, an example and illustrative graph is provided.

Asymptotic independence of bivariate order statistics

It is well known that an extreme order statistic and a central order statistic (os) as well as an intermediate os and a central os from a sample of iid univariate random variables get asymptotically independent as the sample size increases. We extend this result to bivariate random variables, where the os are taken componentwise. An explicit representation of the conditional distribution of bivariate os turns out to be a powerful tool.

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACTIn this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (XT1, X2T, …, XTk, ZT)T = (X11, …, X1n, …, Xk1, …, Xkn, Z1, …, Zn)T and derive the distribution of concomitant of multivariate order statistics arising from X1, X2, …, Xk. 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.

Joint distribution of new sample rank of bivariate order statistics

Let , be independent copies of bivariate random vector with joint cumulative distribution function and probability density function . For , the vector of order statistics of and , respectively, is denoted by . Let , , be a new sample from , which is independent from . Let be the rank of order statistics in a new sample and be the rank of order statistics in a new sample . We derive the joint distribution of discrete random vector and a general scheme wherein the distributions of new and old samples are different is considered. Numerical examples for given well-known distribution are also provided.

On Bivariate Order Statistics from Elliptical Distributions

In this article, by considering a (m + n)-dimensional random vector (XT, YT)T = (X1, …, Xm, Y1, …, Yn)T having a multivariate elliptical distribution, and denoting X(m) = (X(1), …, X(m))T and Y(n) = (Y(1), …, Y(n))T for the vectors of order statistics arising from X and Y, respectively, we derive the exact joint distribution of (X(r), Y(s))T, for r = 1, …, m and s = 1, …, n, and also joint distribution of (aTX(m), bTY(n))T where and . Further, by considering an elliptical distribution for the (m + n + 1)-dimensional random vector (X0, XT, YT)T, and treating X0 as a covariate variable, we present mixture representations for joint distributions of (X0, X(m), Y(n))T and (X0, aTX(m), bTY(n))T in terms of multivariate unified skew-elliptical distributions. These mixture representations enable us to obtain the best predictors of X0 based on X(m) and Y(n), and X0 based on aTX(m) and bTY(n), and so on. Finally, we illustrate the usefulness of our results by a real-life data.

Reliability and Mean Residual Life of Complex Systems With Two Dependent Components Per Element

The reliability and mean residual life of complex systems are discussed. These systems consist of n elements each having two s -dependent subcomponents. The reliability of such systems involves the distributions of bivariate order statistics, and are connected with a bivariate binomial distribution. The mean residual life function of complex systems with intact components at time t is also discussed. Some examples and graphical representations are given.

Some novel discrete distributions under fourfold sampling schemes and conditional bivariate order statistics

This paper presents some novel trivariate discrete distributions that are obtained by modifying the bivariate binomial distribution. These distributions are important probability models for the development of conditional bivariate order statistics. The distributional properties of bivariate order statistics are studied and derived under the condition that certain values of the underlying random vectors (X,Y) are truncated and fall in the threshold set {(t,s)∈R2:t≤u,s≤v}, (u,v)∈R2.

Dependence structure of bivariate order statistics with applications to Bayramoglu’s distributions

We study the dependence structure of bivariate order statistics from bivariate distributions, and prove that if the underlying bivariate distribution H is positive quadrant dependent (PQD) then so is each pair of bivariate order statistics. As an application, we show that if H is PQD, the bivariate distribution K+(n), recently proposed by Bairamov and Bayramoglu (2012) [1], is greater than or equal to Baker’s (2008) [2] distribution H+(n), and hence K+(n) attains a correlation higher than that of H+(n). We give two explicit forms of the intractable K+(n) and prove that for all n≥2, K+(n) is PQD regardless of H. We also show that if H is PQD, K+(n) converges weakly to the Fréchet–Hoeffding upper bound as n tends to infinity.

A General Recurrence Relation for the Moments of Bivariate and Trivariate Order Statistics

In this article, by using the dropping argument, a general recurrence relation satisfied by the joint cumulative distribution functions of order statistics from any arbitrary bivariate distribution function is established. This recurrence relation is the first bivariate version of the basic triangle rule for order statistics arisen from univariate distribution function. Finally, this relation is extended to the trivariate case. These lead to similar identities for product moments (of any order) of order statistics.

On the computing of the moments of bivariate order statistics

In this paper, we establish general recurrence relations satisfied by the product moments (of any order) of bivariate order statistics from any arbitrary bivariate uniform distribution function. Moreover, we present formulae to easily compute the product moments (of any order) of bivariate order statistics from any arbitrary bivariate distribution function, with positive left endpoints, or with negative right endpoints.

The Asymptotic Distribution Theory of Bivariate Order Statistics

In this paper the limit distribution function (d.f.) of general bivariate order statistics (o.s.) (extreme, intermediate and central) is studied by the notion of the exceedances of levels and characteristic function (c.f.) technique. The advantage of this approach is to give a simple and unified method to derive the limit d.f. of any bivariate o.s. The conditions under which the limit d.f. splits into the product of the limit marginals are obtained. Some illustrative examples are given.

Estimating the covariance of bivariate order statistics with applications

In this note we develop a new analytic bootstrap-type method for estimating the covariance between the bivariate order statistics X r: n and Y s: n . Correlation and regression estimators follow straightforward from this result. An easy-to-use method for calculating the nonparametric percentile confidence regions for bivariate quantiles is developed and illustrated using biological data. A specific case of interest is the application to bivariate medians, i.e., correlated marginal medians.

On Moments of Bivariate Order Statistics

In the present paper, we give the exact explicit expression for the product moments (of any order) of bivariate order statistics (o.s.) from any arbitrary continuous bivariate distribution function (d.f.). Furthermore, for any arbitrary bivariate uniform d.f., universal distribution-free bounds for the differences of any two different product moments (of order (1,1) or (-1,1)) are given.

On the Estimation of a Support Curve of Indeterminate Sharpness

We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.