Comparing Fisher information in record values and their concomitants with random observations

In a sequence of bivariate random variables {(X _i , Y _i ), i≥1} from a continuous distribution with a real parameter θ, general comparison results between the amount of Fisher information about θ contained in the sequence of the first n records and their concomitants, and the desired information in an i.i.d. sample of size n from the parent distribution are established. Some relationships between reliability properties and the proposed criteria are obtained in situations in which the univariate counterpart of the underlying bivariate family belongs to location, scale or shape families. It is also shown that in some classes of bivariate families, the concerned information property is equivalent to that of its univariate counterpart. The proposed procedure is illustrated by considering several examples.     

Fisher Information in Record Values and Their Concomitants: A Comparison of Two Sampling Schemes

Two sampling designs via inverse sampling for generating record data and their concomitants are considered: single sample and multisample. The purpose here is to compare the Fisher information in these two sampling schemes. It is shown that the comparison criterion depends on the underlying distribution. Several general results are established for some parametric families and their well known subclasses such as location-scale and shape families, exponential family and proportional (reversed) hazard model. Farlie-Gumbel-Morgenstern (FGM) family, bivariate normal distribution, and some other common bivariate distributions are considered as examples for illustrations and are classified according to this criterion.     

Application of Concomitants of Order Statistics of Independent Non Identically Distributed Random Variables in Estimation

In this article, we obtain expressions for the pdf of a single concomitant of order statistic and the joint pdf of a pair of concomitants of order statistics of independent non identically distributed random variables. Using these expressions, we find the means, variances and covariances of order statistics arising from independent non identically distributed bivariate Pareto distributions. A method of estimation of a common parameter involved in several bivariate Pareto distributions using concomitants of order statistics is also 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.     

New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics

We consider a generalization of the bivariate Farlie-Gumbel-Morgenstern (FGM) distribution by introducing additional parameters. For the generalized FGM distribution, the admissible range of the association parameter allowing positive quadrant dependence property is shown. Distributional properties of concomitants for this generalized FGM distribution are studied. Recurrence relations between moments of concomitants are presented.     

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.     

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.     

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.     

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.     

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.     

An introduction to copula-based bivariate reliability Concepts

Several attempts were made in the literature to generalize the notions based on univariate quantiles to higher dimensions. As quantile-based reliability concepts are receiving much attention, it is important to address these problems in the field of Reliability theory. In this paper, bivariate reliability concepts using the dependence structure are introduced. The properties and characterizations of the bivariate reliability concepts are presented; it includes the characterization based on the relationship between bivariate hazard rate and bivariate mean residual life. The bivariate reliability concepts in reversed time are also studied.     

Distribution of extremes of<Emphasis Type="Italic">r</Emphasis><Superscript>th</Superscript> concomitant from the Morgenstern family

In the present paper the distribution theory of maximum and minimum of ther ^th concomitants from k independent subgroups each of same size m from the Morgenstern family is investigated. Some applications of the results in estimation of the scale parameter of a marginal variable in the bivariate uniform distribution and a selection problem are discussed.     

Importance sampling to evaluate bivariate probabilities with an application to sequential Phase II trials

In this paper we propose an importance sampling method for evaluating probabilities associated with the margins of a 2 × 2 table in a fixed or multistage design. These probabilities arise in the work of Conaway and Petroni (1994) on sequential phase I1 trials with bivariate endpoints. The importance sampling method proposed in this paper provides an efficient method, relative to exact enumeration and Monte Carlo techniques with no variance reduction, of estimating these probabilities. The method can also be adapted to estimate probabilities for the designs of Jennison and Turnbull (1993).     

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.     

On additive-multiplicative hazards model

In survival analysis and reliability theory, a fundamental problem is the study of lifetime properties of a live organism or system. In this regard, there have been considered and studied several models based on different concepts of ageing such as hazard rate and mean residual life. In this paper, we consider an additive-multiplicative hazard model (AMHM) and study some reliability and ageing properties of the proposed model. We then specify the bivariate models whose conditionals satisfy AMHM. Several properties of the proposed bivariate model are investigated and adequacy of the model is evaluated based on a real data set.     

Parameter estimation of Cambanis-type bivariate uniform distribution with Ranked Set Sampling

The concept of ranked set sampling (RSS) is applicable whenever ranking on a set of sampling units can be done easily using a judgment method or based on an auxiliary variable. In this paper, we consider a study variable Y correlated with the auxiliary variable X and use it to rank the sampling units. Further (X,Y) is assumed to have Cambanis-type bivariate uniform (CTBU) distribution. We obtain an unbiased estimator of a scale parameter associated with the study variable Y based on different RSS schemes. We perform the efficiency comparison of the proposed estimators numerically. We present the trends in the efficiency performance of estimators under various RSS schemes with respect to parameters through line and surface plots. Further, we develop a Matlab function to simulate data from CTBU distribution and present the performance of proposed estimators through a simulation study. The results developed are implemented to real-life data also.KEYWORDS: Ranked set sampling, concomitants of order statistics, Cambanis-type bivariate uniform distribution, best linear unbiased estimatorSUBJECT CLASSIFICATIONS: 62D05, 62F07, 62G30     

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.     

Bounds for expectations of concomitants

We present sharp bounds on expected values of concomitants based on a sample of identically distributed random pairs. The dependence between pair components is described by regression functions or modelled by copulas, or generated by sampling without replacement from finite populations.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.