Bivariate Family Research Articles

A New Method of Generating Truncated Bivariate Families of Distributions

The modeling of complex data has attracted several researchers for the quest of generating new probability distributions. The joint modeling of two variables asks for some additional complexities as a bivariate distribution is needed. The field of research in developing bivariate families of distributions is somewhat new. In certain situations, the domain of data is restricted and some truncated distribution is required. Several univariate truncated families of distributions are available for modeling of a single variable but the bivariate truncated families of distributions has not been studied and in this paper, we have proposed a new bivariate truncated families of distributions. A specific sub-family has been proposed by using the bivariate Burr as a base-line distribution, resulting in a bivariate truncated Burr family of distributions. Some important statistical properties of the proposed family has been studied, which include the marginal and conditional distributions, bivariate reliability, and bivariate hazard rate functions. The maximum likelihood estimation for the parameters of the family is also carried out. The proposed bivariate truncated Burr family of distributions is studied for the Burr baseline distributions, giving rise to the bivariate truncated Burr-Burr distribution. The new bivariate truncated Burr-Burr distribution is explored in detail and several statistical properties of the new distribution are studied, which include the marginal and conditional distributions, product, ratio, and conditional moments. The maximum likelihood estimation for the parameters of the proposed distribution is done. The proposed bivariate truncated Burr-Burr distribution is used to model some real data sets. It is found that the proposed distribution performs better than the other distributions considered in this study.

Construction of New Bivariate Semilinear Copulas Based on the Ali–Mikhail–Haq, Farlie–Gumbel–Morgenstern and Plackett Families

Copulas are quite a popular, flexible, and useful tool for multivariate modeling in many fields of applications, including finance and insurance, actuarial sciences, biomedical studies, geostatistics, and hydrology. In this paper, we use the concept of diagonal function to build new bivariate copulas based on the well known and important Ali–Mikhail–Haq (AMH), Farlie–Gumbel–Morgenstern (FGM) and Plackett families. We also study some dependence properties of the newly constructed semilinear copulas, namely, the Spearman’s [Formula: see text] coefficient. In particular, we find that the range of this rank correlation coefficient varies between weak and moderate values for the transformed AMH and FGM families, although it varies between moderate and strong values for the transformed Plackett families. Moreover, in all cases, the Spearman’s [Formula: see text] coefficient reaches negative and positive values. Finally, we present a synthetic data generation algorithm, as well as a classical method of estimation of the copula association parameter [Formula: see text] for each of the six transformed bivariate families.

Some Tsallis entropy measures in concomitants of generalized order statistics under iterated FGM bivariate distribution

<p>Shannon differential entropy is extensively applied in the literature as a measure of dispersion or uncertainty. Nonetheless, there are other measurements, such as the cumulative residual Tsallis entropy (CRTE), that reveal interesting effects in several fields. Motivated by this, we study and compute Tsallis measures for the concomitants of the generalized order statistics (CGOS) from the iterated Farlie-Gumbel-Morgenstern (IFGM) bivariate family. Some newly introduced information measures are also being considered for CGOS within the framework of the IFGM family, including Tsallis entropy, CRTE, and an alternative measure of CRTE of order $ \eta $. Applications of these results are given for order statistics and record values with uniform, exponential, and power marginals distributions. In addition, the empirical cumulative Tsallis entropy is suggested as a method to calculate the new information measure. Finally, a real-world data set has been analyzed for illustrative purposes, and the performance is quite satisfactory.</p>

A New Bivariate Family Based on Archimedean Copulas: Simulation, Regression Model and Application

A New Bivariate Family Based on Archimedean Copulas: Simulation, Regression Model and Application

Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications

In this article, a new family of bivariate discrete distributions is proposed based on the copula concept, in the so-called bivariate discrete odd generalized exponential-G family. Some distributional properties, including the joint probability mass function, joint survival function, joint failure rate function, median correlation coefficient, and conditional expectation, are derived. After proposing the general class, one special model of the new bivariate family is discussed in detail. The maximum likelihood approach is utilized to estimate the family parameters. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood estimators. Finally, the importance of the new bivariate family is explained by means of two distinctive real data sets in various fields.

New bivariate family of distributions based on any copula function: Statistical properties

In this paper, new bivariate family of distributions based on any copula is established. Of this, we introduce new bivariate Topp-Leone family based on Farlie-Gumbel-Morgenstern (FGM) copula. As a special case, we concentrate our study in the new bivariate Topp-Leone-Exponential-Exponential (BFGMTLEE) distribution based on FGM copula. Some of its properties are developed such as product moments, moment generating function and entropy.

A novel bivariate Lomax-G family of distributions: Properties, inference, and applications to environmental, medical, and computer science data

<abstract><p>This paper presents a novel family of bivariate continuous Lomax generators known as the BFGMLG family, which is constructed using univariate Lomax generator (LG) families and the Farlie Gumbel Morgenstern (FGM) copula. We have derived several structural statistical properties of our proposed bivariate family, such as marginals, conditional distribution, conditional expectation, product moments, moment generating function, correlation, reliability function, and hazard rate function. The paper also introduces four special submodels of the new family based on the Weibull, exponential, Pareto, and log-logistic baseline distributions. The study establishes metrics for local dependency and examines the significant characteristics of the proposed bivariate model. To provide greater flexibility, a multivariate version of the continuous FGMLG family are suggested. Bayesian and maximum likelihood methods are employed to estimate the model parameters, and a Monte Carlo simulation evaluates the performance of the proposed bivariate family. Finally, the practical application of the proposed bivariate family is demonstrated through the analysis of four data sets.</p></abstract>

Bayesian inference for the parameters of mortality rate in the models of dependent lives with application in life insurance

In this paper, the Bayesian inference of the model of dependent lives is considered. We use the bivariate Gompertz (BGP) distribution. This new bivariate survival model and its extension have recently been proposed by Shoaee and Khorram. This model is more flexible and can be applied in the actuarial science of life insurance, and reliability theory in competing risks and shock models. As we know, the maximum likelihood estimates do not always exist and therefore cannot always be calculated. So, the Bayesian estimations are considered using the squared error loss function and a priori distributions that create a dependency between the hyper-parameters. But given the assumptions, one can see that explicit expressions cannot be obtained for Bayesian estimations. The importance sampling method is proposed to calculate the Bayes estimations and also to create the corresponding HPD credible intervals of the unknown parameters. The Monte Carlo simulation studies and analysis of a real data set are also performed to evaluate the proposed method for estimating parameters. Finally, a generalization of the dependent lives models is presented and the Bayesian estimate for this new bivariate family distribution is introduced and a comparison is made between the models of this new family.

Information measures for order statistics and their concomitants from Cambanis bivariate family

One of the most efficient and robust methods in estimation is to find the Fisher information matrix (FIM). In this paper, the FIM concerning order statistics (OSs) and their concomitants about the shape-parameter vector of Cambanis bivariate distribution is derived. This method contains information conveyed by singly or multiply censored bivariate samples drawn from the Cambanis bivariate distribution. In addition, we obtain and study the Fisher information (FI) about the mean and shape parameters in the concomitants of OSs for this bivariate distribution based on exponential and power function distributions. Finally, the information entropy of Shannon in the OSs and their concomitants based on the Cambanis distribution is studied. Some numerical computations were performed on the FI which lend support to the theoretical results.

A New Flexible Univariate and Bivariate Family of Distributions for Unit Interval (0, 1)

We propose a new generator for unit interval which is used to establish univariate and bivariate families of distributions. The univariate family can serve as an alternate to the Kumaraswamy-G univariate family proposed earlier by Cordeiro and de-Castro in 2011. Further, the new generator can also be used to develop more alternate univariate and bivariate G-classes such as beta-G, McDonald-G, Topp-Leone-G, Marshall-Olkin-G and Transmuted-G for support (0, 1). Some structural properties of the univariate family are derived and the estimation of parameters is dealt. The properties of a special model of this new univariate family called a New Kumaraswamy-Weibull (NKwW) distribution are obtained and parameter estimation is considered. A Monte Carlo simulation is reported to assess NKwW model parameters. The bivariate extension of the family is proposed and the estimation of parameters is described. The simulation study is also conducted for bivariate model. Finally, the usefulness of the univariate NKwW model is illustrated empirically by means of three real-life data sets on Air Conditioned Failures, Flood and Breaking Strength of Fibers, and one real-life data on UEFA Champion’s League for bivariate model.

Some bivariate and multivariate families of distributions: Theory, inference and application

<abstract> <p>The bivariate and multivariate probability distributions are useful in joint modeling of several random variables. The development of bivariate and multivariate distributions is relatively tedious as compared with the development of univariate distributions. In this paper we have proposed a new method of developing bivariate and multivariate families of distributions from the univariate marginals. The properties of the proposed families of distributions have been studies. These properties include marginal and conditional distributions; product, ratio and conditional moments; joint reliability function and dependence measures. Statistical inference about the proposed families of distributions has also been done. The proposed bivariate family of distributions has been studied for Weibull baseline distribution giving rise to a new bivariate Weibull distribution. The properties of the proposed bivariate Weibull distribution have been studied alongside maximum likelihood estimation of the unknown parameters. The proposed bivariate Weibull distribution has been used for modeling of real bivariate data sets and we have found that the proposed bivariate Weibull distribution has been a suitable choice for the modeling of data used.</p> </abstract>

Generalization of the Ratio of Minimized Kullback- Leibler Divergence Discrimination Technique to Bivariate Marshall-Olkin Family

Generalization of the Ratio of Minimized Kullback- Leibler Divergence Discrimination Technique to Bivariate Marshall-Olkin Family

Relationship between Psychological Status and Health Behaviors during the Coronavirus Disease Pandemic in Japanese Community-Dwelling Older Adults.

The coronavirus disease (COVID-19) continues to be a widespread pandemic. We investigated the relationship between anxiety/stress and health behaviors during the COVID-19 pandemic in homebound Japanese older adults during January and February 2021. We surveyed 1507 community-dwelling, older Japanese adults using a self-administered questionnaire on primary attributes, including family structure, evaluation of psychological anxiety/stress, and health behaviors. Participants were divided into four anxiety/stress groups based on the frequency of experiencing anxiety/stress, and their association with health behaviors was analyzed using bivariate and multivariate analyses. Responses were received from 469 (31.1%) respondents. In the bivariate analysis, age and family structure were significantly associated with anxiety/stress (p < 0.01). The health behaviors significantly associated with anxiety/stress were walking, balanced eating habits, limited snacking, regular lifestyle, and dental visits. Logistic regression analysis was performed using the variables in the bivariate analysis that showed a significant association with anxiety/stress status as independent variables. Finally, age and dietary habits were significantly associated with anxiety/stress status. No significant associations were found between any other variables. Among older adults living in the rural areas of Japan during the COVID-19 pandemic, anxiety/stress status was significantly associated with age and dietary habits but not with other health behaviors.

Bivariate Koornwinder\u2013Sobolev Orthogonal Polynomials

The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function rho (t) such that rho (t)^2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.

Location charts based on concomitants of ranked set samples from Morgenstern family

This paper surveys the location chart based on concomitants of ranked set sampling (RSS), a triangular modification of RSS, which is known as maximum ranked set sampling with unequal sample (MRSSU) and the extreme ranked set sampling (ERSS) from the Farlie–Gumbel–Morgenstern bivariate family. We also remarked both heavy-tailed symmetric and skewed distributions throughout the paper. For the performance evaluation of the charts, extensive utilization of the Monte Carlo simulations is carried out using average run length (ARL), extra quadratic loss (EQL), relative ARL (RARL), and performance comparison index (PCI) measures, and finally, conclusion ends this research.

Bivariate Weibull-G Family Based on Copula Function: Properties, Bayesian and non-Bayesian Estimation and Applications

This paper aims to obtain a new flexible bivariate generalized family of distributions based on FGM copula, which is called bivariate FGM Weibull-G family. Some of its statistical properties are studied as marginal distributions, product moments, and moment generating functions. Some dependence measures as Kendall’s tau and median regression model are discussed. After introducing the general class, four special sub models of the new family are introduced by taking the baseline distributions as Pareto, inverted Topp-Leone, exponential, and Rayleigh distributions. Maximum likelihood and Bayesian approaches are used to estimate the model unknown parameters. Further, percentile bootstrap confidence interval and bootstrap-t confidence interval are estimated for the model’s parameters. A Monte-Carlo simulation study is carried out of the maximum likelihood and Bayesian estimators. Finally, we illustrate the importance of the proposed bivariate family using two real data sets in medical field.

The General Tail Dependence Function in the Marshall-Olkin and Other Parametric Copula Models with an Application to Financial Time Series

The accurate understanding of the dependence structure implied by the parametric models studied in statistical and financial literature has drawn growing attention in recent times. In particular, tail dependence is crucial in this analysis. We study the general tail dependence function in some of the most common copula models found in literature, in which this function has not been obtained. These models are used by statisticians and practitioners alike. With the general tail dependence function, we cover positive and non-positive tail dependence often overlooked. The use of the general tail dependence function generalises the well known approach of using the survival copula to tackle upper tail dependence. We present relevant results regarding tail dependence related functions. We include a broad guide for bivariate families and Hierarchical Archimedean copulas in dimensions 3 and 4. In the multivariate case we study the Marshall-Olkin copula, examples of models based on Max-Id distributions and Extreme Value copulas among others. In an empirical section we exemplify with real data the usefulness of our results by modelling arbitrary types of tail dependence. Furthermore we show how our approach can yield better estimates of Value at Risk than other standard approaches.

A Class of Bivariate Semiparametric Families of Distributions

The study of semiparametric families is useful because it provides methods of extending families for adding flexibility in fitting data. The main aim of this paper is to introduce a class of bivariate semiparametric families of distributions. One especial bivariate family of the introduced semiparametric families is discussed in details with its sub-models and different properties. In most of the cases the joint probability distribution, joint distribution and joint hazard functions can be expressed in compact forms. The maximum likelihood and Bayesian estimation are considered for the vector of the unknown parameters. For illustrative purposes a data set has been re-analyzed and the performances are quite satisfactory. A simulation study is performed to see the performances of the estimators.

The Bivariate Transmuted Family of Distributions: Theory and Applications

The bivariate distributions are useful in simultaneous modeling of two random variables. These distributions provide a way to model models. The bivariate families of distributions are not much widely explored and in this article a new family of bivariate distributions is proposed. The new family will extend the univariate transmuted family of distributions and will be helpful in modeling complex joint phenomenon. Statistical properties of the new family of distributions are explored which include marginal and conditional distributions, conditional moments, product and ratio moments, bivariate reliability and bivariate hazard rate functions. The maximum likelihood estimation (MLE) for parameters of the family is also carried out. The proposed bivariate family of distributions is studied for the Weibull baseline distributions giving rise to bivariate transmuted Weibull (BTW) distribution. The new bivariate transmuted Weibull distribution is explored in detail. Statistical properties of the new BTW distribution are studied which include the marginal and conditional distributions, product, ratio and conditional momenst. The hazard rate function of the BTW distribution is obtained. Parameter estimation of the BTW distribution is also done. Finally, real data application of the BTW distribution is given. It is observed that the proposed BTW distribution is a suitable fit for the data used.

A Generator of Bivariate Distributions: Properties, Estimation, and Applications

In 2020, El-Morshedy et al. introduced a bivariate extension of the Burr type X generator (BBX-G) of distributions, and Muhammed presented a bivariate generalized inverted Kumaraswamy (BGIK) distribution. In this paper, we propose a more flexible generator of bivariate distributions based on the maximization process from an arbitrary three-dimensional baseline distribution vector, which is of interest for maintenance and stress models, and expands the BBX-G and BGIK distributions, among others. This proposed generator allows one to generate new bivariate distributions by combining non-identically distributed baseline components. The bivariate distributions belonging to the proposed family have a singular part due to the latent component which makes them suitable for modeling two-dimensional data sets with ties. Several distributional and stochastic properties are studied for such bivariate models, as well as for its marginals, conditional distributions, and order statistics. Furthermore, we analyze its copula representation and some related association measures. The EM algorithm is proposed to compute the maximum likelihood estimations of the unknown parameters, which is illustrated by using two particular distributions of this bivariate family for modeling two real data sets.