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.

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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.

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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**.

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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.

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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.

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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.

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Multivariate order statistics especially **Bivariate** order statistics have attracted the interest of several researchers, for example, see [1]. The distribution of **bivariate** order statistics can be easily obtained from the **bivariate** binomial distribution, which was first introduced by [2]. Considering a **bivariate** sample, David et al. [3] studied the distribution of the sam- ple rank for a concomitant of an order statistic. Bairamove and Kemalbay [4] 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 [5] 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 [6] used the derived jpdf by [5] to derive the joint distribution on new sample rank of **bivariate** order statistics. Moreover, Barakat [7] 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 [8].

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Various Class Weibull G **distributions** have been discussed such as Weibull Pareto distribution by Alzaatreh, et al. [2].Copulas are a general tool to construct multivariate **distributions** and measure the dependence structure between random variables. The paper of Abd elaal [1] 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.

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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.

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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.

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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).

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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. [2] introduced a method which based on the choice of pairs of order statistics of the marginal **distributions**. [8] studied on construction of continuous **bivariate** **distributions** that possesses the Positive Quadrant Dependence property. [13] introduced a generalization of Farlie-Gumbel-Morgentern (FGM) distribution family. They extend the maximal correlation coefficient for FGM family. Furthermore, [14] introduced **bivariate** and multivariate generalization of quadratic transmutation distribution family. Proposal of [14] 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**

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For the univariate case, the LT power mixture (LTPM) operator as introduced by Abate and Whitt [4] (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 [5] 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 [3] (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 [6] (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.

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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

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, 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.

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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).

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An interesting relatively-recent development in EDAs is the use of copula tech- niques (see [17] 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 [40].

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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.

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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

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A more general approach is taken by Bennett and Kennedy (2004), who use copulas in conjunction with a triangular no-arbitrage condition to price quanto FX options, i.e. FX options whose payout is in a third currency. Similar to Bikos and Taylor and Wang, they use option-implied densities as margins for the **bivariate** distribution. However, they estimate their copula function by fitting an entire set of option contracts in the third bilateral (over different strike prices) instead of fitting just the implied correlation coefficient. This additional information enables them to use a Gaussian copula which is perturbed by a cubic spline and which therefore allows for a more flexible dependence structure between the three currency pairs. In the context of the quanto pricing problem this approach is appealing because the perturbation function indicates the extent of departure from the standard Black Scholes model corresponding to a joint lognormal distribution.

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