Preservation of dependence concepts under bivariate weighted distributions

ABSTRACT

In this paper, we provide conditions under which some bivariate dependence structures are preserved under bivariate weighted distributions. Bivariate weighted distributions whose dependence structure is the same as the original distribution are characterized. Finally, we discuss some examples to show the usefulness of our results.     

A strategy for constructing multivariate distributions

Given a random vector (X₁,…, X_n) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented     

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.     

On the bivariate weighted exponential distribution based on the generalized exponential distribution

In this paper, we introduce a bivariate weighted exponential distribution based on the generalized exponential distribution. This class of distributions generalizes the bivariate distribution with weighted exponential marginals (BWE). We derive different properties of this new distribution. It is a four-parameter distribution, and the maximum-likelihood estimator of unknown parameters cannot be obtained in explicit forms. One data set has been re-analyzed and it is observed that the proposed distribution provides better fit than the BWE distribution.     

On a class of bivariate mixed Sarmanov distributions

Multivariate distributions are more and more used to model the dependence encountered in many fields. However, classical multivariate distributions can be restrictive by their nature, while Sarmanov's multivariate distribution, by joining different marginals in a flexible and tractable dependence structure, often provides a valuable alternative. In this paper, we introduce some bivariate mixed Sarmanov distributions with the purpose to extend the class of bivariate Sarmanov distributions and to obtain new dependency structures. Special attention is paid to the bivariate mixed Sarmanov distribution with Poisson marginals and, in particular, to the resulting bivariate Sarmanov distributions with negative binomial and with Poisson‐inverse Gaussian marginals; these particular types of mixed distributions have possible applications in, for example modelling bivariate count data. The extension to higher dimensions is also discussed. Moreover, concerning the dependency structure, we also present some correlation formulas.     

Linear hypothesis testing for weighted functional data with applications

In socioeconomic areas, functional observations may be collected with weights, called weighted functional data. In this paper, we deal with a general linear hypothesis testing (GLHT) problem in the framework of functional analysis of variance with weighted functional data. With weights taken into account, we obtain unbiased and consistent estimators of the group mean and covariance functions. For the GLHT problem, we obtain a pointwise F-test statistic and build two global tests, respectively, via integrating the pointwise F-test statistic or taking its supremum over an interval of interest. The asymptotic distributions of test statistics under the null and some local alternatives are derived. Methods for approximating their null distributions are discussed. An application of the proposed methods to density function data is also presented. Intensive simulation studies and two real data examples show that the proposed tests outperform the existing competitors substantially in terms of size control and power.     

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.     

Aspects of the mean residual life order for weighted distributions

In this paper, we first provide conditions for preservation of the mean residual life (mrl) order under weighting. Then we apply the obtained results to establish our results about preservation of the decreasing mrl class by weighted distributions. In addition, we present some results for comparing the original random variable to its weighted version in terms of the mrl order. Also, some examples are given to illustrate the results.     

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.     

Weighted Marshall–Olkin bivariate exponential distribution

Recently, Gupta and Kundu [R.D. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (2009), pp. 621–634] have introduced a new class of weighted exponential (WE) distributions, and this can be used quite effectively to model lifetime data. In this paper, we introduce a new class of weighted Marshall–Olkin bivariate exponential distributions. This new singular distribution has univariate WE marginals. We study different properties of the proposed model. There are four parameters in this model and the maximum-likelihood estimators (MLEs) of the unknown parameters cannot be obtained in explicit forms. We need to solve a four-dimensional optimization problem to compute the MLEs. One data set has been analysed for illustrative purposes and finally we propose some generalization of the proposed model.     

On multivariate weighted distributions

The concept of weighted distributions is well-known in the literature concerning observational studies and surveys in research related to forestry, ecology, bio-medicine and many other areas (cf. Rao (1965)). This paper extends the idea of weighted distributions to multivariate case. A few multivariate orderings have been defined and some partial ordering results are presented. Some results regarding multivariate positive and negative dependence are also discussed. Multivariate weighted distributions - joint, marginal and conditional have been defined and some important results concerning them are presented along with an illustration. The Multivariate Poisson Negative Hypergeometric Distribution has been derived.     

On the reciprocity of connections in weighted and unweighted networks

Bivariate Bernoulli and bivariate geometric distributions are proposed to model reciprocity in weighted and unweighted networks. Sampling properties of commonly used reciprocity measures are studied, under both presence and absence of reciprocity. Extensions to situations where link parameters depend on outgoing and incoming nodes are investigated. Results are illustrated with a data set, on 50 anesthesia providers at a hospital, that describes the number of times an anesthesia provider starting an operation transfers his/her responsibilities to another anesthesia provider.     

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.     

On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.     

Inference for a Hidden Truncated (Both-Sided) Bivariate Pareto (II) Distribution

The Pareto distribution is a simple model for non negative data with a power law probability tail. Income and wealth data are typically modeled using some variant of the classical Pareto distribution. In practice, it is frequently likely that the observed data have been truncated with respect to some unobserved covariable. In this paper, a hidden truncation formulation of this scenario is proposed and analyzed. A bivariate Pareto (II) distribution is assumed for the variable of interest and the unobserved covariable. Distributional properties of the resulting model are investigated. A variety of parameter estimation strategies (under the classical set up) are investigated.     

Non parametric analysis of gap times for bivariate recurrent event data with cure fraction

Recurrent event data often arise in longitudinal studies. In many applications, subjects may experience two different types of events alternatively over time or a pair of subjects may experience recurrent events of the same type. Medical advances have made it possible for some patients to be cured such that the disease of interest does not recur. In this article, we consider non parametric analysis of bivariate recurrent event data with cure fraction. Using the inverse-probability weighted (IPW) approach, we propose non parametric estimators for the proportion of cured patients and for the joint distribution functions of bivariate recurrence times of the uncured ones. The asymptotic properties of the proposed estimators are established. Simulation study indicates that the proposed estimators perform well in finite samples.     

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.     

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.     

An asymmetric multivariate weibull distribution

Abstract

A class of multivariate laws as an extension of univariate Weibull distribution is presented. A well known representation of the asymmetric univariate Laplace distribution is used as the starting point. This new family of distributions exhibits some similarities to the multivariate normal distribution. Properties of this class of distributions are explored including moments, correlations, densities and simulation algorithms. The distribution is applied to model bivariate exchange rate data. The fit of the proposed model seems remarkably good. Parameters are estimated and a bootstrap study performed to assess the accuracy of the estimators.