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.     

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.     

Concomitants of generalized order statistics from Farlie–Gumbel–Morgenstern distributions

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. Here, we consider concomitants of generalized order statistics for the Farlie–Gumbel–Morgenstern bivariate distributions and study recurrence relations between their moments. We derive the joint distribution of concomitants of two generalized order statistics and obtain their product moments. Application of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results.     

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.     

On a characteristic property of generalized Pareto distributions, extreme value distributions and their max domains of attraction

Using the independence of an arbitrary random variable Y and the weighted minima of independent, identically distributed random variables with weights depending on Y, we characterize extreme value distributions and generalized Pareto distributions. A discussion is made about an analogous characterization for distributions in the max domains of attraction of extreme value limit laws.     

Bayesian inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.     

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.     

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.     

Parameter estimation in nonlinear regression models 1

Fisher information contained in record values, inter-record times and their concomitants from a sample of fixed size is derived in general and explicit expressions are deduced for some specific known bivariate classes of distributions. A comparison between fixed sampling and inverse sampling schemes with equal number of records and concomitants is also carried out. We also consider parameter estimation based on bivariate records and a small simulation study is done.     

Estimation of parameter of Morgenstern type bivariate exponential distribution using concomitants of order statistics

In this paper we have considered concomitants of order statistics arising from Morgenstern type bivariate exponential distribution and their applications in estimating the unknown parameter involved in the distribution. We have obtained the best linear unbiased estimator of a parameter involved in Morgenstern type bivariate exponential distribution using both complete and censored samples.     

SHARP BOUNDS FOR EXPECTATIONS OF kTH RECORD INCREMENTS

This paper presents sharp bounds for expectations of non‐adjacent increments of kth record statistics, measured in various scale units, for a sequence of independent identically distributed random variables with continuous cumulative distribution function. The results for kth record spacings are considered as special cases. The paper also characterizes probability distributions for which the bounds are attained.     

Characterization of the geometric distribution using properties of order statistics

Using certain properties of order statistics, the geometric distribution has been characterized when the components are independent and identically distributed. When the components are independent, the geometric distribution has been characterized in the class of either IFR or DFR discrete distributions. In particular, Ferguson's (1967) characterization theorem for independent components in a sample of size two has been extended in several directions.     

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.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Order statistics and their concomitants from multivariate normal mean–variance mixture distributions with application to Swiss Markets Data

In this article, by considering a multivariate normal mean–variance mixture distribution, we derive the exact joint distribution of linear combinations of order statistics and their concomitants. From this general result, we then deduce the exact marginal and conditional distributions of order statistics and their concomitants arising from this distribution. We finally illustrate the usefulness of these results by using a Swiss markets dataset.     

General properties and concomitants of Morgenstern bivariate gamma distribution and their applications in estimation

In the present article, the general distribution theory of Morgenstern type bivariate gamma distribution and the properties of the concomitants of order statistics from it are discussed. Estimation for the parameters of the distribution using the concomitants and method of moments are proposed and their properties are presented.     

Current records and record range with some applications

In a sequence of independent and identically distributed (iid) random variables, the upper (lower) current records and record range are studied. We derive general recurrence relations between the single and product moments for the upper and lower current records based on Weibull and positive Weibull distributions, as well as Pareto and negative Pareto distributions, respectively. Moreover, some asymptotic results for general current records are established. In addition, a recurrence relation and an explicit formula for the moments of record range based on the exponential distribution are given. Finally, numerical examples are presented to illustrate and corroborate theoretical results.     

Concomitants of k-record values arising from Morgenstern family of distributions and their applications in parameter estimation

In this paper, we have obtained the marginal and joint distributions of concomitants of k-record values for the Morgenstern family of distributions (MFD) and hence obtained the moments and product moments of concomitants of k-record values. Applying this results we have derived the best linear unbiased estimators of some parameters involved in Morgenstern type bivariate logistic distribution which belongs to MFD based on concomitants of k-record values.     

A note on moments of order statistics from exchangeable variates

Joshi (1973) and Balakrishnan and Malik (1985) have derived some some identities for the moments of order statistics from independent and identically distributed random variables. In this paper, we make use of a basic result due to David and Joshi (1968) and show that these identities for the moments also hold when the order statistics arise from exchangeable variables.     

On the distribution of the maximum of bivariate normal random variables with general means and variances

A great amount of effort has been devoted to achieving exact expressions for moments of order statistics of independent normal random variables, as well as the dependent case with the same correlation coefficients, means and variances. It does not seem as if there are handy formulae for the order statistics of even the simple bivariate normal random variables when the means and variances are allowed to be different. In this paper we give an explicit formula for the Lanl ace-Stielties Transform of the maximum of bivariate normal random variables by which we obtain formulae for the first two moments in the standard way.