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.     

On Pitman's measure of closeness of k-records

Pitman closeness of both the upper and lower k-record statistics to the population quantiles of a location–scale family of distributions is studied. For the population median, the Pitman-closest k-record is also determined. In the case of symmetric distributions, the Pitman closeness probabilities of k-record statistics are shown to be distribution-free, and explicit expressions are also derived for these probabilities. Exact expressions are derived for the required probabilities for uniform and exponential distributions. Numerical results are given for these families and also the Pitman-closest k-record is determined.     

Simultaneous Closeness of k-Records

In this article, we study the simultaneous Pitman closeness of upper (and lower) k-records to a common parameter of the parent distribution and obtain general expressions for the corresponding probabilities. Since the usual record values are contained in the k-records, the corresponding results for the usual records can be deduced as special cases. The results are then applied to location-scale families with an emphasis on population quantiles. Exact expressions are derived for some common distributions such as exponential and uniform. In each case, the simultaneous closest k-record to a population quantile is determined.     

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.     

Equal-tailed and shortest Bayesian tolerance intervals based on exponential k-records

The problem of constructing the equal-tailed and shortest Bayesian tolerance intervals that control percentages in both tails of the exponential distribution based on k-record values is considered. Equal-tailed and shortest Bayesian tolerance factors are derived. Practical examples using real and simulated k-record values are given to illustrate the proposed results.     

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.     

Lower bounds on the expectations of upper record values

For infinite sequences of independent random variables with identical continuous distributions, we establish optimal lower bounds on the deviations of the expectations of record values from population means in units generated by the central absolute moments of various orders. The bounds are non-negative for the classic record values, and non-positive for the other kth records with k?2. We also provide analogous bounds for the record increments.     

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.     

On k-th record times,record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.     

On Moments of k-Records from Discrete Distributions

In this article, we establish recurrence relations satisfied by first and second moments of k-records arising from discrete distributions. Then we use these relations to obtain means, variances, and correlation coefficients of geometric k-records. We consider all the three known types of k-records: strong, ordinary, and weak.     

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.     

An Estimator of the Exponent of Regular Variation Based on K-Record Values

Let {X _n:n ≥ 1} be an i.i.d. sequence of random variables with a continuous distribution function F. Under the assumption that the upper tail of Fis regularly varying with exponent 1/α, α > 0, we study the asymptotic properties of an estimator of α based on k-record values.     

Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.     

The Distribution of the Record Position and its Applications

This is mainly an expository article on the positions of records in sequences of ordered elements. Such sequences are obtained, for example, when observing and ordering continuous iid random variables. In practice, records are of interest, for example, in meteorology and sports. A k-record is obtained when a new element is placed at position k counted from the top. The sequences of time points, when new k-records occur, are studied by elementary random walk methods. In the last section, it is shown that the time scale can be changed so that the time points of the k-records follow, approximately, a Poisson process.     

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.     

Log-gamma-generated families of distributions

We introduce two new general families of continuous distributions, generated by a distribution F and two positive real parameters α and β which control the skewness and tail weight of the distribution. The construction is motivated by the distribution of k-record statistics and can be derived by applying the inverse probability integral transformation to the log-gamma distribution. The introduced families are suitable for modelling the data with a significantly skewed and heavy-tailed distribution. Various properties of the introduced families are studied and a number of estimations and data fitness on real data are given to illustrate the results.     

Rank-based empirical likelihood inference on medians of k populations

We propose a nonparametric method, called rank-based empirical likelihood (REL), for making inferences on medians and cumulative distribution functions (CDFs) of k populations. The standard distribution-free approach to testing the equality of k medians requires that the k population distributions have the same shape. Our REL-ratio (RELR) test for this problem requires fewer assumptions and can effectively use the symmetry information when the distributions are symmetric. Furthermore, our RELR statistic does not require estimation of variance, and achieves asymptotic pivotalness implicitly. When the k populations have equal medians we show that the REL method produces valid inferences for the common median and CDFs of k populations. Simulation results show that the REL approach works remarkably well in finite samples. A real data example is used to illustrate the proposed REL method.     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.     

Chi-squared tests in kth-moment sufficient dimension reduction

We present a novel approach to sufficient dimension reduction for the conditional kth moments in regression. The approach provides a computationally feasible test for the dimension of the central kth-moment subspace. In addition, we can test predictor effects without assuming any models. All test statistics proposed in the novel approach have asymptotic chi-squared distributions.