The Lack of Memory Property in the Density Form for the Bivariate Setup

The lack of memory property is a characterizing property of the exponential distribution in the continuous domain. In the bivariate setup different generalizations of the same are available in terms of survival function. We extend this lack of memory property in terms of bivariate probability density function and examine its characterization properties. In this process the density version of the lack of memory property can be interlinked with conditionally specified exponential distribution, bivariate reciprocal coordinate subtangent of the density curve and a few other derived measures.     

Bivariate extension of (dynamic) cumulative past entropy

Recently, the concept of cumulative residual entropy (CRE) has been studied by many researchers in higher dimensions. In this article, we extend the definition of (dynamic) cumulative past entropy (DCPE), a dual measure of (dynamic) CRE, to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure.     

Bivariate extension of generalized cumulative past entropy

The cumulative past entropy (CPE) of order α, a dual measure of cumulative residual entropy (CRE) of order α, has recently been proposed as a suitable extension of CPE. In this article, we extend the definition of (dynamic) CPE of order α (DCPE(α)) to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE(α) for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE(α) are obtained for conditional distributions. Along with some characterization results it is shown that the proposed generalized measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure and prove interrelation with some existing stochastic orders.     

Process capability index for bivariate exponentially distributed quality characteristics and its sampling properties

In this paper, a process capability index for two correlated quality characteristics jointly following bivariate exponential distribution has been proposed. The expectation and sampling variance of the estimated index have been derived. Choice of the natural process interval corresponding to a specified coverage probability has been discussed.     

Multivariate Birnbaum–Saunders distribution: properties and associated inference

The univariate fatigue life distribution proposed by Birnbaum and Saunders [A new family of life distributions. J Appl Probab. 1969;6:319–327] has been used quite effectively to model times to failure for materials subject to fatigue and for modelling lifetime data and reliability problems. In this article, we introduce a Birnbaum–Saunders (BS) distribution in the multivariate setting. The new multivariate model arises in the context of conditionally specified distributions. The proposed multivariate model is an absolutely continuous distribution whose marginals are univariate BS distributions. General properties of the multivariate BS distribution are derived and the estimation of the unknown parameters by maximum likelihood is discussed. Further, the Fisher's information matrix is determined. Applications to real data of the proposed multivariate distribution are provided for illustrative purposes.     

Test of fit for Marshall–Olkin distributions with applications

Marshall and Olkin [1967. A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44], introduced a bivariate distribution with exponential marginals, which generalizes the simple case of a bivariate random variable with independent exponential components. The distribution is popular under the name ‘Marshall–Olkin distribution’, and has been extended to the multivariate case. L2-type statistics are constructed for testing the composite null hypothesis of the Marshall–Olkin distribution with unspecified parameters. The test statistics utilize the empirical Laplace transform with consistently estimated parameters. Asymptotic properties pertaining to the null distribution of the test statistic and the consistency of the test are investigated. Theoretical results are accompanied by a simulation study, and real-data applications.     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

A new bivariate distribution with weighted exponential marginals and its multivariate generalization

Gupta and Kundu (Statistics 43:621–643, 2009) recently introduced a new class of weighted exponential distribution. It is observed that the proposed weighted exponential distribution is very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the weighted exponential marginals. Different properties of this new bivariate distribution have been investigated. This new family has three unknown parameters, and it is observed that the maximum likelihood estimators of the unknown parameters can be obtained by solving a one-dimensional optimization procedure. We obtain the asymptotic distribution of the maximum likelihood estimators. Small simulation experiments have been performed to see the behavior of the maximum likelihood estimators, and one data analysis has been presented for illustrative purposes. Finally we discuss the multivariate generalization of the proposed model.     

A simple nonparametric test for bivariate symmetry about a line

Over the years many researchers have dealt with testing the hypotheses of symmetry in univariate and multivariate distributions in the parametric and nonparametric setup. In a multivariate setup, there are several formulations of symmetry, for example, symmetry about an axis, joint symmetry, marginal symmetry, radial symmetry, symmetry about a known point, spherical symmetry, and elliptical symmetry among others. In this paper, for the bivariate case, we formulate a concept of symmetry about a straight line passing through the origin in a plane and accordingly develop a simple nonparametric test for testing the hypothesis of symmetry about a straight line. The proposed test is based on a measure of deviance between observed counts of bivariate samples in suitably defined pairs of sets. The exact null distribution and non-null distribution, for specified classes of alternatives, of the test statistics are obtained. The null distribution is tabulated for sample size from n=5 up to n=30. The null mean, null variance and the asymptotic null distributions of the proposed test statistics are also obtained. The empirical power of the proposed test is evaluated by simulating samples from the suitable class of bivariate distributions. The empirical findings suggest that the test performs reasonably well against various classes of asymmetric bivariate distributions. Further, it is advocated that the basic idea developed in this work can be easily adopted to test the hypotheses of exchangeability of bivariate random variables and also bivariate symmetry about a given axis which have been considered by several authors in the past.     

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.     

Functionally Compatible Local Characteristics for the Local Specification of Priors in Graphical Models

Abstract.  The local specification of priors in non-decomposable graphical models does not necessarily yield a proper joint prior for all the parameters of the model. Using results concerning general exponential families with cuts, we derive specific results for the multivariate Gamma distribution (conjugate prior for Poisson counts) and the Wishart distribution (conjugate prior for Gaussian models). These results link the existence of a locally specified joint prior to the solvability of a related marginal problem over the cliques of the graph.     

On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

Form-invariance under weighted sampling

In this paper, form-invariant weighted distributions are considered in an exponential family. The class of bivariate distribution with invariant property is identified under exponential weight function. The class includes some of the custom bivariate models. The form-invariant multivariate normal distributions are obtained under a quadratic weight function.     

Absolute continuous bivariate generalized exponential distribution

Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.     

Computing the Point-biserial Correlation under Any Underlying Continuous Distribution

The connection between the point-biserial and biserial correlations is well-established when the underlying distribution is bivariate normal. For many other bivariate distributions, the formula that links these two quantities is not straightforward to derive or does not have a closed form. We propose a simple technique that enables researchers to compute one of these correlations when the other is specified. For this, we take advantage of the constancy of their ratio, which can be easily approximated for any distribution. We illustrate the proposed method using several examples and discuss its extension to the ordinal case. We believe that this approach is potentially useful in stochastic simulation..     

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.     

Nonnormality of Linear Combinations of Normal Random Variables

Hamedani and Tata (1975) showed that the bivariate normal distribution is determined uniquely by any countably infinite collection of distinct linear combinations of the variables and by no finite number of them. It is shown here that this characterization of bivariate normal distribution cannot be extended to the multivariate case. More specifically, it is shown that the multivariate normality of subsets (r < n) of the normal variables X ₁, X ₂, …, X_n together with the normality of an infinite number of linear combinations of them do not guarantee the joint normality of these variables.     

A class of absolutely continuous bivariate distributions

Block and Basu bivariate exponential distribution is one of the most popular absolutely continuous bivariate distributions. Extensive work has been done on the Block and Basu bivariate exponential model over the past several decades. Interestingly it is observed that the Block and Basu bivariate exponential model can be extended to the Weibull model also. We call this new model as the Block and Basu bivariate Weibull model. We consider different properties of the Block and Basu bivariate Weibull model. The Block and Basu bivariate Weibull model has four unknown parameters and the maximum likelihood estimators cannot be obtained in closed form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. We propose to use the EM algorithm for computing the maximum likelihood estimators of the unknown parameters. The proposed EM algorithm can be carried out by solving one non-linear equation at each EM step. Our method can be also used to compute the maximum likelihood estimators for the Block and Basu bivariate exponential model. One data analysis has been preformed for illustrative purpose.     

Truncation invariant dependence structures

In this paper, a special class of m-dimensional distribution functions which can be uniquely determined in terms of their 2-dimensional marginals is studied. The members of the class can be characterized as having truncation invariant dependence structure. The representation given in this paper provides a physical meaning to the multivariate Cook-Johnson distribution, and introduces a systematic way of generating higher dimensional distributions by using rich 2-dimensional distributions provided that the 2-dimensional marginals are compatible. A class of 3-dimensional multivariate normal distribution has been generated and bounds in terms of lower dimensional marginals are provided.     

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.