# bivariate distributions

## Top PDF bivariate distributions: ### Characterization of Bivariate Distributions Using Concomitants of Generalized (k) Record Values

The study of record concomitants was initiated by Houchens (1984). Develop- ments in the theory and applications of concomitants of record values and concomi- tants of order statistics open the door for analysis of data arising from bivariate distributions in a new perspective. For a description about the theory of concomi- tants of record values, see Ahsanullah and Nevzorov (2000). For a recent account on the use of concomitants of record values in estimation, see Chacko (2007), Chacko and Thomas (2006, 2008). Veena and Thomas (2008) and Thomas and Veena (2011) have attempted for the ﬁrst time to characterize some bivariate dis- tributions using the properties of concomitants of order statistics. Also in the available literature it seems Thomas and Veena (2014) is the only paper in which some results on characterizing a class of bivariate distributions by properties of concomitants of record values are discussed. ### Assessment of Performance of Correlation Estimates in Discrete Bivariate Distributions using Bootstrap Methodology

Little attention has been given to the correlation coefficient when data come from discrete or continuous non-normal populations. In this paper we considered the efficiency of two correlation coefficients which are from the same family, Pearson’s and Spearman’s estimators. Two discrete bivariate distributions were examined, the Poisson and the Negative Binomial. The comparison between these two estimators took place using classical and bootstrap techniques for the construction of confidence intervals. Thus, these techniques are also subject to comparison. Simulation studies were also used for the relative efficiency and bias of the two estimators. Pearson’s estimator performed slightly better than Spearman’s. ### Recent developments on the construction of bivariate distributions with fixed marginals

Section 2 is a review of some basic properties of the bivariate distributions with fixed marginals. Then, in Sections 3 to 6, we will focus on the recent developments on the FGM- related distributions, including Sarmanov and Lee’s distributions, Baker’s distributions and Bayramoglu’s distributions. Some new results are provided. This complements the most recent works mentioned above. Finally, in Section 7 we briefly discuss some other related distributions. ### ON GENERALIZED SARMANOV BIVARIATE DISTRIBUTIONS

In this paper, we construct classes of bivariate distributions which are generalizations of Sarmanov and Sarmanov-Lee models. These distributions have a simple analytical form like the FGM models and, as in the “normal” case, the correlation coefficient ρ totally governs the dependence between the variables. Some dependence properties of these distributions are also discussed. ### The Cambanis family of bivariate distributions: Properties and applications

The Cambanis family of bivariate distributions was introduced as a generalization of the Farlie-Gumbel-Morgenstern system. The present work is an attempt to investigate the distributional characteristics and applications of the family. We derive various co- eﬃcients of association, dependence concepts and time-dependent measures. Bivariate reliability functions such as hazard rates and mean residual life functions are analysed. The application of the family as a model for bivariate lifetime data is also demonstrated. ### A class of continuous bivariate distributions with linear sum of hazard gradient components

The rules established in Theorem 4 may serve as a useful guide for constructing bivariate distributions possessing property (6). The building scheme may be relaxed under additional available information regarding monotone behavior of marginal failure rates. In fact, the class of bivariate distributions L(x; a) may have arbitrary combina- tion of marginal failure rates: increasing, decreasing, constant, bathtub, etc., implying corresponding restrictions for the parameter space, of course. ### On the joint distribution of order statistics from independent non-identical bivariate distributions

Multivariate order statistics especially Bivariate order statistics have attracted the interest of several researchers, for example, see . The distribution of bivariate order statistics can be easily obtained from the bivariate binomial distribution, which was first introduced by . Considering a bivariate sample, David et al.  studied the distribution of the sam- ple rank for a concomitant of an order statistic. Bairamove and Kemalbay  introduced new modifications of bivariate binomial distribution, which can be applied to derive the distribution of bivariate order statistics if a certain number of observations are within the given threshold set. Barakat  derived the exact explicit expression for the product moments (of any order) of bivariate order statistics from any arbitrary continuous bivari- ate distribution function (df ). Bairamove and Kemalbay  used the derived jpdf by  to derive the joint distribution on new sample rank of bivariate order statistics. Moreover, Barakat  studied the limit behavior of the extreme order statistics arising from n two- dimensional independent and non-identically distributed random vectors. The class of limit dfs of multivariate order statistics from independent and identical random vectors with random sample size was fully characterized by . ### Manuscript Title & Authors

Various Class Weibull G distributions have been discussed such as Weibull Pareto distribution by Alzaatreh, et al. .Copulas are a general tool to construct multivariate distributions and measure the dependence structure between random variables. The paper of Abd elaal  provided several methods of constructing bivariate distributions with copula functions.The main aim of this article is to introduce bivariate Weibull C h i s q u a r e (BWCH) model based on the most used copula function named Gaussian copula with a suitable organization. The paper is organized as follows. Section 2presents the bivariateWeibull Chi-square (BWCH) model based on Gaussian copula function. The maximum likelihood estimates (MLEs) for the model parameters are demonstrated in Section 3. In Section 4, the flexibility of the model is explained. Finally, the performance of the suggested model using a simulation data is discussed Section 5. ### Bivariate beta-generated distributions with applications to well-being data

The class of beta-generated distributions (Commun. Stat. Theory Methods 31:497–512, 2002; TEST 13:1–43, 2004) has received a lot of attention in the last years. In this paper, three new classes of bivariate beta-generated distributions are proposed. These classes are constructed using three different definitions of bivariate distributions with classical beta marginals and different covariance structures. We work with the bivariate beta distributions proposed in (J. Educ. Stat. 7:271–294, 1982; Metrika 54:215–231, 2001; Stat. Probability Lett. 62:407–412, 2003) for the first proposal, in (Stat. Methods Appl. 18: 465–481, 2009) for the second proposal and (J. Multivariate Anal. 102:1194–1202, 2011) for the third one. In each of these three classes, the main properties are studied. Some specific bivariate beta-generated distributions are studied. Finally, some empirical applications with well-being data are presented. ### Idealized models of the joint probability distribution of wind speeds

The two probability density functions (pdfs) we will con- sider, the bivariate Rice and Weibull distributions, both start with simple assumptions regarding the distributions of the wind components. The bivariate Rice distribution follows di- rectly from the assumption of Gaussian components with isotropic variance, but nonzero mean. In contrast, the bi- variate Weibull distribution is obtained from nonlinear trans- formations of the magnitudes of Gaussian, isotropic, mean- zero components. While the univariate Weibull distribution has been found to generally be a better fit to observed wind speeds than the univariate Rice distribution (particularly over the oceans, e.g. Monahan, 2006, 2007), the direct connec- tion of the Rice distribution to the distribution of the compo- nents (which the Weibull distribution does not have) is useful from a modelling and theoretical perspective (e.g. Cakmur et al., 2004; Monahan, 2012a; Culver and Monahan, 2013; Sun and Monahan, 2013; Drobinski et al., 2015). The six- parameter bivariate Rice distribution that we will consider is more flexible than the five-parameter bivariate Weibull dis- tribution, and able to model a broader range of dependence structures. Furthermore, it is directly connected to the uni- variate distributions and dependence structure of the wind components. However, the bivariate Weibull distribution is mathematically much simpler than the bivariate Rice distri- bution and easier to use in practice. Other flexible bivariate distributions for non-negative random variables exist, such as the α-µ distribution discussed in Yacoub (2007). Because the Weibull and Rice distributions are common models for the univariate wind speed distributions, this study will focus specifically on their bivariate generalizations. ### A note on finding peakedness in bivariate normal distribution using Mathematica

In bivariate distributions, the property of peakedness is important and has two impending problems i.e. the marginal distributions corresponding to a bivariate distribution would have peakedness (i) separately and (ii) jointly with relationship parameter, usually given by the linear correlation coefficient. Kurtosis is one such property which has not been studied for a bivariate model according to the theory developed by Horn (1983). Quraishi and Haq (1999) and Hussain et al. (2000) modified the Horn’s measure for bivariate discrete probability distributions and bivariate normal distribution respectively. In this article Horn’s measure of peakedness is modified for the bivariate normal distribution. It is also shown that computer algebra system Mathematica can easily be used in computation for peakedness and plotting its contour (Wolfram, 1991). ### Construction of Bivariate Distribution by Mixing Positively Dependent and Negatively Dependent Distributions

Univariate models are insufficient to explain random phenomena. Today, data such as drought, wind speed and rainfall are measured together with the variables that may affect them. With the development of technology, the construction of continuous bivariate distribution functions with given marginals has become an importance. When creating new bivariate distributions, models that can express high correlation are generally tried to be obtained.  introduced a method which based on the choice of pairs of order statistics of the marginal distributions.  studied on construction of continuous bivariate distributions that possesses the Positive Quadrant Dependence property.  introduced a generalization of Farlie-Gumbel-Morgentern (FGM) distribution family. They extend the maximal correlation coefficient for FGM family. Furthermore,  introduced bivariate and multivariate generalization of quadratic transmutation distribution family. Proposal of  draws our attention in particular. Because the transition from univariate case to bivariate or multivariate cases is not so easy. While in univariate case the real line is the complement of the probabilities, at least in the bivariate ### Maximum Entropy Empirical Likelihood Methods Based on Bivariate Laplace Transforms and Moment Generating Functions

For the univariate case, the LT power mixture (LTPM) operator as introduced by Abate and Whitt  (pp. 92-93) for the univariate set up for creating new distri- bution using Laplace transforms (LT) can be used to generate many distributions with closed form LT. Furthermore, it can also be used to generate distribution which is infinitely divisible. In this paper we shall generalize the LTPM to a bivariate version, the BLTPM operator and show that the BLTPM operator can be used to generate new bivariate distributions with closed from BLT. Furthermore, the traditional survival copulas such as the Clayton copula, see Shih and Louis  can be used as a LT copula if the property of complete monotonicity of two specific related functions are satisfied. For another class of survival copulas, see Crowder  (pp. 121-138). Consequently, some distribution or survival function copulas which are defined using a generator based on a LT of a nonnegative random variable can be used to generate new distribution with prescribed marginals specified by their marginal LTs. A similar bivariate PM operator based on distribution functions or survival functions, the BDSPM operator has been introduced by Marshall and Olkin  (pp. 834-836) to create new distributions functions and with frailty induced distributions, it also related to a class of distribution or survival Copula functions defined with a generator. The BLTPM operator can be used to generate bivariate infinite divisible distributions with closed form BLTs. It appears to be useful to have bivariate infinite divisible distributions for joint modeling as they are related to corresponding bivariate Lévy processes with stationary and inde- pendent increments. These types of processes are useful as they often lead to elegant results in risk theory in actuarial sciences and in finance. ### A family of bivariate Pareto distributions

Since the bivariate distributions in the proposed family have identical marginal distributions, in modelling and analysis of data, a crucial aspect that diﬀerentiate them in a practical situation is the diﬀerences in the dependence or association between the constituent random variables. Thus a study of various dependence concepts and measures become crucial when discussing family properties, as they tell us the extent to which the variables are associated and also the nature of their relationships. There are three distinct approaches in the study of association. The ﬁrst one is through numerical measures like the Pearson’s correlation coeﬃcient, the Kendall’s tau, Spearman’s rho, Gini’s measure and Blomqvist’s β. Presently we discuss the correlation coeﬃcient and postpone the study of the other measures in a seperate work when the copulas of the member distributions are taken up. A second approach is to study the dependence properties. The six basic properties of positive dependence are (1) total positivity of order 2 (2) stochastic increase (3) right tail increase (4) positive association (5) positive quadrant dependence and (6) positive correlation or Cov(X, Y ) ≥ 0. Negative dependence properties are deﬁned as the dual’s of these. Among the six properties, the relative stringency is expressed as follows ### Copulas for finance

, and so density contours are ellipsoids. They play an important role in finance. In figure 13, we have plotted the contours of bivariate density for the gaussian copula and differents marginal distributions. We verify that the gaussian copula with two gaussian marginals correspond to the bivariate gaussian distribution, and that the contours are ellipsoids. Building multivariate distributions with copulas becomes very easy. For example, figure 13 contains two other bivariate densities with different margins. In figure 14, margins are the same, but we use a copula of the Frank family. For each figure, we have choosen the copula parameter in order to have the same Kendall’s tau (τ = 0.5). The dependence structure of the four bivariate distributions can be compared. ### Characterizations of a family of bivariate Pareto distributions

dullness property. The family (1) includes various known and unknown bivariate distributions. The members of the family can be derived by choosing appropriate univariate distribution of Z. Table 1 provides some members of family according to diﬀerent choices of the distribution of Z. It may be noted that the proper- ties of ¯ F (x, y) can be inferred from the corresponding properties of ¯ G(z). The members of the family listed in Table 1, include bivariate Pareto with indepen- dent marginals, Mardia’s(1962) Type 1 model and bivariate Burr distribution. For more properties one could refer to Sankaran et al. (2014). ### DICE: A New Family of Bivariate Estimation of Distribution Algorithms based on Dichotomised Multivariate Gaussian Distributions

An interesting relatively-recent development in EDAs is the use of copula tech- niques (see  for a survey). Copulas are a statistical tool that allow a mul- tivariate dependency to be decomposed into a univariate marginal distribution function and a copula, which describes the dependence structure between the variables. Both aspects can then be modelled separately. This can allow partic- ular EDA models to be applied to a wider set of problem domains. For example, copulas might allow a Gaussian distribution model to be used even when the problem univariate marginal distribution itself is not Gaussian. One application of copulas techniques to bivariate EDAs was the development of a more general copula-based version of the MIMIC algorithm in . ### Information theoretic measures of dependence, compactness, and non-gaussianity for multivariate probability distributions

Abstract. A basic task of exploratory data analysis is the characterisation of “structure” in multivariate datasets. For bivariate Gaussian distributions, natural measures of depen- dence (the predictive relationship between individual vari- ables) and compactness (the degree of concentration of the probability density function (pdf) around a low-dimensional axis) are respectively provided by ordinary least-squares re- gression and Principal Component Analysis. This study con- siders general measures of structure for non-Gaussian distri- butions and demonstrates that these can be defined in terms of the information theoretic “distance” (as measured by rela- tive entropy) between the given pdf and an appropriate “un- structured” pdf. The measure of dependence, mutual infor- mation, is well-known; it is shown that this is not a useful measure of compactness because it is not invariant under an orthogonal rotation of the variables. An appropriate rotation- ally invariant compactness measure is defined and shown to reduce to the equivalent PCA measure for bivariate Gaus- sian distributions. This compactness measure is shown to be naturally related to a standard information theoretic measure of non-Gaussianity. Finally, straightforward geometric inter- pretations of each of these measures in terms of “effective volume” of the pdf are presented. ### A New Multivariate Weibull Distribution

Record statistics has been studied by several authors for various probability distributions. Ahsanullah (1992) has provided general results for distribution of records for continuous probability distributions. A comprehensive review of record statistics can be found in Ahsanullah (1995) and Nevzorov (2001). Often we have a sample from some bivariate distribution and the sample is arranged with respect to one of the variable. The automatically shuffled other variable is called concomitant of ordered variable. The concomitant of records appear when the bivariate sample is arranged with respect to records. The density function of nth concomitant of record is given by Ahsanullah (1995) as 