Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.     

Testing for bivariate gumbel against bivariate new better than used in expectation

It is shown that a bivariate survival function is both New Better than Used in Expectation (NBUE) and New Worse than Used in Expectation (NWUE) if and only if it is a bivariate Gumbel distribution. Statistical procedures are then presented to test whether that, within the class of bi-variate NBUE survival functions, a survival function is a Gumbel's bivariate exponential.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

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.     

Some bivariate uniform distributions

Bivariate uniform distributions with dependent components are readily derived by distribution function transformations of the components of non-uniform dependent continuous bivariate random variables (X,Y). Contour plots of joint density functions show the various, and varying, forms of dependence which can arise from different distributional forms for (X,Y) and aids the choice of bivariate uniform distributions as empirical models.     

Properties and applications of the sarmanov family of bivariate distributions

We discuss properties of the bivariate family of distributions introduced by Sarmanov (1966). It is shown that correlation coefficients of this family of distributions have wider range than those of the Farlie-Gumbel-Morgenstern distributins. Possible applications of this family of bivariate distributions as prior distributins in Bayesian inference are discussed. The density of the bivariate Sarmanov distributions with beta marginals can be expressed as a linear combination of products of independent beta densities. This pseudoconjugate property greatly reduces the complexity of posterior computations when this bivariate beta distribution is used as a prior. Multivariate extensions are derived.     

Bivariate generalized gamma distributions of Kibble's type

In this paper, a new type of bivariate generalized gamma (BGG) distribution derived from the bivariate gamma distribution of Kibble [Two-variate gamma-type distribution. Sankh?a 1941;5:137–150] by means of a power transformation is presented. The explicit expressions of statistical properties of the BGG distribution are presented. The estimation of marginal and dependence parameters using the method of moments and the method of inference functions for margins are discussed, and their performance through a Monte Carlo simulation study is assessed. Finally, an example is given to illustrate the applicability of the distributions introduced here.     

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.     

The noncentral bivariate chisquared distribution and extensions

The distribution of certain correlated noncentral chisquared variates P, Q, is termed the noncentral bivariate chisquared distribution. Moment generating functions of the distributions of (P, Q), (P+Q) and other quadratic forms have been obtained. A relationship to the linear case of the noncentral Wishart distribution is indicated. Convolution properties and applications are presented.     

An approximation to percentiles of a variable of the bivariate normal distribution when the other variable is truncated,with applications

A Cornish-Fisher expansion is used to approximate the per-centiles of a variable of the bivariate normal distribution when the other variable is truncated. The expression is in terms of the bivariate cumulants of a singly truncated bivariate normal distribution. The percentiles are useful in the problem of personnel selection where we use a screening variable to screen applicants for employment and a correlated performance variable to screen employees for rehiring. This paper provides a bivariate cumulants table for determining the cutoff score of the performance variable. The following two problems are also con¬sidered: (1) determine the proportion of applicants who would have been successful had no screening been applied, and (2) determine the proportion of individuals being rejected byscreening who would have been successful had they been hired, The variable that is used to measure job performance and the variable that measures the outcome of an aptitude test are assumed to be jointly normally distributed with correlation ρ     

A bivariate density estimation method based on level crossings

This article introduces a method of nonparametric bivariate density estimation based on a bivariate sample level crossing function, which leads to the construction of a bivariate level crossing empirical distribution function (BLCEDF). An efficiency function for this BLCEDF relative to the classical empirical distribution function (EDF), is derived. The BLCEDF gives more efficient estimates than the EDF in the tails of any underlying continuous distribution, for both small and large sample sizes. On the basis of BLCEDF we define a bivariate level crossing kernel density estimator (BLCKDE) and study its properties. We apply the BLCEDF and BLCKDE for various distributions and provide results of simulations that confirm the theoretical properties. A real-world example is given.     

Testing whether survival function is bivariate new better than used

We present statistical procedures to test that a life distribution is bivariate exponential against the alternative that it is bivariate new better than used (BNBU).     

Some bivariate families of lagrangian probability distributions

The bivariate Lagrange expansion, given by Poincare (1986), has been explained and slightly modified which gives bivariate Lagrangian probability models. A generalized bivariate Lagrangian Poisson distribution with six parameters has been obtained and studied. Also, the bivariate Lagrangian binomial, bivariate Lagrangian negative binomial and bivariate Lagrangian logarithmic series distribution have been obtained.     

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution 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.     

Independent or dependent competing risks: does it make a difference

This article investigates the consequences of departures from independence when the component lifetimes in a series system are exponentially distributed. Such departures are studied when the joint distribution is assumed to follow either one of the three Gumbel bivariate exponential models, the Downton bivariate exponential model, or the Oakes bivariate exponential model. Two distinct situations are considered. First, in theoretical modeling of series systems, when the distribution of the component lifetimes is assumed, one wishes to compute system reliability and mean system life. Second, errors in parametric and nonparametric estimation of component reliability and component mean life are studied based on life-test data collected on series systems when the assumption of independence is made     

Forms for the distribution of ss ilkptxcxtt statistic for bivariate sphericity

This paper examines the nonnuU distribution of the ellipticity statistic used for testing sphericity in a bivariate normal population. The distribution function is expressed as mixtures of F distribution functions, where the mixing probabilities are binomial and mgatiire binomial. A simple approximation to the distribution is suggested and examined     

Fisher's Information on the Correlation Coefficient in Bivariate Logistic Models

From a theoretical perspective, the paper considers the properties of the maximum likelihood estimator of the correlation coefficient, principally regarding precision, in various types of bivariate model which are popular in the applied literature. The models are: 'Full-Full', in which both variables are fully observed; 'Censored-Censored', in which both of the variables are censored at zero; and finally, 'Binary-Binary', in which both variables are observed only in sign. For analytical convenience, the underlying bivariate distribution which is assumed in each of these cases is the bivariate logistic. A central issue is the extent to which censoring reduces the level of Fisher's information pertaining to the correlation coefficient, and therefore reduces the precision with which this important parameter can be estimated.     

A backward construction and simulation of correlated Poisson processes

In this paper, we consider a generalisation of the backward simulation method of Duch et al. [New approaches to operational risk modeling. IBM J Res Develop. 2014;58:1–9] to build bivariate Poisson processes with flexible time correlation structures, and to simulate the arrival times of the processes. The proposed backward construction approach uses the Marshall–Olkin bivariate binomial distribution for the conditional law and some well-known families of bivariate copulas for the joint success probability in lieu of the typical conditional independence assumption. The resulting bivariate Poisson process can exhibit various time correlation structures which are commonly observed in real data.     

Mixture formulations of a bivariate negative binomial distribution with applications

This paper considers further mixture formulations of the bivariate negative binomial (BNB) distribution of Edwards and Gurland (1961) and Subrahmaniam (1966). These formulations and some known ones are applied (1) to obtain a bivariate generalized negative binomial (BGNB) distribution of Bhattacharya (1966), (2) to establish a connection between the accident-proneness models given by the BNB, BGNB and Bhattacharya's bivariate distributions, and (3) to compute the grade correlation and distribution function of the Wicksell-Kibble bivariate gamma distribution.     

Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.