A versatile bivariate distribution on a bounded domain: another look at the product moment correlation

The Farlie-Gumbel-Morgenstern (FGM) family has been investigated in detail for various continuous marginals such as Cauchy, normal, exponential, gamma, Weibull, lognormal and others. It has been a popular model for the bivariate distribution with mild dependence. However, bivariate FGMs with continuous marginals on a bounded support discussed in the literature are only those with uniform or power marginals. In this paper we study the bivariate FGM family with marginals given by the recently proposed two-sided power (TSP) distribution. Since this family of bounded continuous distributions is very flexible, the properties of the FGM family with TSP marginals could serve as an indication of the structure of the FGM distribution with arbitrary marginals defined on a compact set. A remarkable stability of the correlation between the marginals has been observed.     

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.     

Bivariate beta regression models: joint modeling of the mean,dispersion and association parameters

In this paper a bivariate beta regression model with joint modeling of the mean and dispersion parameters is proposed, defining the bivariate beta distribution from Farlie–Gumbel–Morgenstern (FGM) copulas. This model, that can be generalized using other copulas, is a good alternative to analyze non-independent pairs of proportions and can be fitted applying standard Markov chain Monte Carlo methods. Results of two applications of the proposed model in the analysis of structural and real data set are included.     

A new extension of the FGM copula for negative association

In this paper we introduced a single parameter, absolutely continuous and radially symmetric bivariate extension of the Farlie-Gumbel-Morgenstern (FGM) family of copulas. Specifically, this extension measures the higher negative dependencies than most FGM extensions available in literature. Closed-form formulas for distribution, quantile, density, conditional distribution, regression, Spearman's rho, Kendall's tau, and Gini's gamma are obtained. In addition, a formula for random variate generations is presented in closed-form to facilitate simulation studies. We conduct both paired and multiple comparisons with Frank, Gaussian, and Plackett copulas to investigate the performance based on Vuong's test. Furthermore, the new copula is compared with Frank, Gaussian, and Plackett copulas using both Kolmogorov-Smirnov and Cramér-von Mises type test statistics. Finally, a bivariate dataset is analyzed to compare and illustrate the flexibility of the new copula for negative dependence.     

Estimation of a common mean vector in bivariate meta-analysis under the FGM copula

We propose a bivariate Farlie–Gumbel–Morgenstern (FGM) copula model for bivariate meta-analysis, and develop a maximum likelihood estimator for the common mean vector. With the aid of novel mathematical identities for the FGM copula, we derive the expression of the Fisher information matrix. We also derive an approximation formula for the Fisher information matrix, which is accurate and easy to compute. Based on the theory of independent but not identically distributed (i.n.i.d.) samples, we examine the asymptotic properties of the estimator. Simulation studies are given to demonstrate the performance of the proposed method, and a real data analysis is provided to illustrate the method.     

Hazard Rate Properties of Systems with Two Components

In this article, we study the hazard rate ordering of lifetimes of two-component systems (series and parallel) by considering some bivariate distributions for the joint distribution of component lifetimes. Such that, we aim to investigate the lifetimes of systems with stochastically dependent and with stochastically independent components, the lifetimes of the components, and a stochastic ordering relation between these lifetimes. In addition to these, the mononotonicity of the hazard rates of the parallel and the series systems for the bivariate Farlie–Gumbel–Morgenstern (FGM) family is studied.     

An investigation of Kendall's τ modified for censored data with applications

To accommodate testing for independence in bivariate data subject to censoring, several modifications of Kendall's τ are discussed. An extensive computer simulation is done to investigate power properties of these modifications under alternatives of the bivariate normal or bivariate exponential types. The statistics are then applied to available heart pacemaker patient survival data.     

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.     

A likelihood ratio test and its modifications for the homogeneity of the covariance matrices of dependent multivariate normals

For the problem of testing the homogeneity of the covariance matrices of several dependent multivariate normals, a likelihood ratio test is derived in this paper. The software SAS can be used to perform the test and the code is given and explained. Several modifications of the test that allow their distributions to be better approximated by the chi-square distribution are also considered. Formulae for calculating approximate sample size and power are derived. Small sample performances of these tests in the case of two dependent bivariate normals are compared to each other and to the competing tests by simulating levels of significance and powers, and recommendation is made of the ones that have good performance. The recommended tests are then applied to real data from a crossover bioequivalence trial.     

A family of bivariate binomial distributions generated by extreme bernoulli distributions

In this paper, bivariate binomial distributions generated by extreme bivariate Bernoulli distributions are obtained and studied. Representation of the bivariate binomial distribution generated by a convex combination of extreme bivariate Bernoulli distributions as a mixture of distributions in the class of bivariate binomial distribution generated by extreme bivariate Bernoulli distribution is obtained. A subfamily of bivariate binomial distributions exhibiting the property of positive and negative dependence is constructed. Some results on positive dependence notions as it relates to the bivariate binomial distribution generated by extreme bivariate Bernoulli distribution and a linear combination of such distributions are obtained.     

ESTIMATING THE COMMON MEAN OF A BIVARIATE NORMAL POPULATION

For estimating the common mean of a bivariate normal distribution, Krishnamoorthy & Rohatgi (1989) proposed some estimators which dominate the maximum likelihood estimator in a large region of the parameter space. We consider some modifications of these estimators and study their risk performance.     

A bivariate extension of the beta generated distribution derived from copulas

In this paper, we introduce a new class of bivariate distributions whose marginals are beta-generated distributions. Copulas are employed to construct this bivariate extension of the beta-generated distributions. It is shown that when Archimedean copulas and convex beta generators are used in generating bivariate distributions, the copulas of the resulting distributions also belong to the Archimedean family. The dependence of the proposed bivariate distributions is examined. Simulation results for beta generators and an application to financial risk management are presented.     

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.     

Basu–Dhar bivariate geometric distribution in the presence of covariates and censored data: a Bayesian approach

In this paper, we introduce classical and Bayesian approaches for the Basu–Dhar bivariate geometric distribution in the presence of covariates and censored data. This distribution is considered for the analysis of bivariate lifetime as an alternative to some existing bivariate lifetime distributions assuming continuous lifetimes as the Block and Basu or Marshall and Olkin bivariate distributions. Maximum likelihood and Bayesian estimators are presented. Two examples are considered to illustrate the proposed methodology: an example with simulated data and an example with medical bivariate lifetime data.     

On the bivariate negative binomial regression model

In this paper, a new bivariate negative binomial regression (BNBR) model allowing any type of correlation is defined and studied. The marginal means of the bivariate model are functions of the explanatory variables. The parameters of the bivariate regression model are estimated by using the maximum likelihood method. Some test statistics including goodness-of-fit are discussed. Two numerical data sets are used to illustrate the techniques. The BNBR model tends to perform better than the bivariate Poisson regression model, but compares well with the bivariate Poisson log-normal regression model.     

Bivariate semi α-Laplace distribution and processes

The Laplace distribution is considered as a better choice for modeling whenever data exhibit high kurtosis and heavier tails than Gaussian tails. Even though this is the case, not much work has been done on bivariate Laplace distribution. In this work, we introduce and study a new class of bivariate distributions called bivariate semi α-Laplace distribution, containing bivariate Laplace distributions. Three characterizations of bivariate semi α-Laplace distribution are obtained. Relation with bivariate semi stable distribution is established. An autoregressive model with bivariate semi α-Laplace marginal distributions is developed.     

Comparisons of some bivariate regression models

The bivariate negative binomial regression (BNBR) and the bivariate Poisson log-normal regression (BPLR) models have been used to describe count data that are over-dispersed. In this paper, a new bivariate generalized Poisson regression (BGPR) model is defined. An advantage of the new regression model over the BNBR and BPLR models is that the BGPR can be used to model bivariate count data with either over-dispersion or under-dispersion. In this paper, we carry out a simulation study to compare the three regression models when the true data-generating process exhibits over-dispersion. In the simulation experiment, we observe that the bivariate generalized Poisson regression model performs better than the bivariate negative binomial regression model and the BPLR model.     

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution. The distributions of the minimum and maximum are obtained as special cases.     

A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.     

Invariant dependence structure under univariate truncation

The class of all bivariate copulas that are invariant under univariate truncation is characterized. To this end, a family of bivariate copulas generated by a real-valued function is introduced. The obtained results are also used in order to show that the Clayton family of copulas (including its limiting elements) coincides with the class of copulas that are invariant under bivariate truncation and contains all exchangeable copulas which are invariant under univariate truncation.