Bivariate Geometric Distribution Research Articles

Negative Binomial and Geometric; Bivariate and Difference Distributions

A similarity and a difference between bivariate negative binomial distribution and bivariate geometric distribution is presented. The distribution of negative binomial difference and geometric difference and the corresponding characteristic function are presented.

Negative Binomial and Geometric; Bivariate and Difference Distributions

A similarity and a difference between bivariate negative binomial distribution and bivariate geometric distribution is presented. The distribution of negative binomial difference and geometric difference and the corresponding characteristic function are presented.

Modeling dragonfly population data with a Bayesian bivariate geometric mixed-effects model

We develop a generalized linear mixed model (GLMM) for bivariate count responses for statistically analyzing dragonfly population data from the Northern Netherlands. The populations of the threatened dragonfly species Aeshna viridis were counted in the years 2015–2018 at 17 different locations (ponds and ditches). Two different widely applied population size measures were used to quantify the population sizes, namely the number of found exoskeletons (‘exuviae’) and the number of spotted egg-laying females were counted. Since both measures (responses) led to many zero counts but also feature very large counts, our GLMM model builds on a zero-inflated bivariate geometric (ZIBGe) distribution, for which we show that it can be easily parameterized in terms of a correlation parameter and its two marginal medians. We model the medians with linear combinations of fixed (environmental covariates) and random (location-specific intercepts) effects. Modeling the medians yields a decreased sensitivity to overly large counts; in particular, in light of growing marginal zero inflation rates. Because of the relatively small sample size (n = 114) we follow a Bayesian modeling approach and use Metropolis-Hastings Markov Chain Monte Carlo (MCMC) simulations for generating posterior samples.

A bivariate geometric distribution via conditional specification: properties and applications

In this article, we discuss a bivariate geometric distribution whose conditionals are geometric distributions and the marginals are not geometric and exhibits negative correlation. Several useful structural properties of the bivariate geometric distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of negative correlation, marginal over-dispersion, distribution of sum, and distribution of the conditional given the sum are also derived. The distribution is shown to be a member of the multi-parameter exponential family and some natural but useful consequences are also outlined. A characterization via conditional failure rates is provided. An extensive simulation study is also carried out to explore the flexibility of the bivariate geometric distribution with various choices of the model parameters. Finally, the distribution is fitted to one bivariate count data set with an inherent negative correlation to illustrate its suitability.

Bivariate lifetime geometric distribution in presence of cure fractions

In this paper, we introduce a Bayesian analysis for bivariate geometric distributions applied to lifetime data in the presence of covariates, censored data and cure fraction using Markov Chain Monte Carlo (MCMC) methods. We show that the use of a discrete bivariate geometric distribution could bring us some computational advantages when compared to standard existing bivariate exponential lifetime distributions introduced in the literature assuming continuous lifetime data as for example, the exponential Block and Basu bivariate distribution. Posterior summaries of interest are obtained using the popular OpenBUGS software. A numerical illustration is introduced considering a medical data set related to the analysis of a diabetic retinopathy data set.

Bivariate Extended Yule Distribution

Here we introduce a bivariate version of the extended Yule distribution (EYD) of Martinez-Rodriguez et al (Computational Statistics and Data Analysis, 2011). The EYD has been found suitable for heavy tailed distributions. For emphasizing the practical relevance of the model, it is shown that the proposed distribution can be viewed as the distribution of the random sum of independently and identically distributed bivariate Bernoulli random variables. It is also observed that the proposed bivariate distribution is a very flexible distribution and it includes the bivariate geometric distribution as its special case. A genesis of the distribution and explicit closed form expressions for its probability mass function, factorial moments and the probability generating function of its conditional distributions are derived. Certain recurrence relations for probabilities, raw moments and factorial moments of the BEYD are also obtained. The method of maximum likelihood estimation is employed for estimating the parameters of the distribution. Some examples of application using data from different fields are included in order to illustrate all these procedures.

Correlations Between Linear and Nonlinear Functions of Discrete Random Variables

In this article correlation between simple linear and nonlinear functions of two random variables have been discussed. The study is carried out for two bivariate geometric distributions.

A bivariate discrete inverse resilience family of distributions with resilience marginals

ABSTRACT In this paper, a new bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distributions, is studied. It is observed that the proposed bivariate distribution is a flexible distribution whose cumulative distribution function has an analytical structure. In addition, a new bivariate geometric distribution can be obtained as a special case. We study different properties of this distribution and propose estimation of its parameters. We will see that the maximum of the variables involved in the proposed bivariate distribution defines some new classes of univariate discrete distributions, which are interesting in their own sake, and can be used to analyze some Reliability systems whose components are positive dependent. Some important futures of this new univariate family of discrete distributions are also studied in details. In addition, a general class of bivariate discrete distributions, whose marginals are exponentiated discrete distributions, is introduced. Moreover, the analysis of two real bivariate data sets is performed to indicate the effectiveness of the proposed models. Finally, we conclude the paper.

On a discrete-time risk model with time-dependent claims and impulsive dividend payments

A discrete-time risk model with a mathematically tractable dependence structure between interclaim times and claim sizes is considered in the presence of an impulsive dividend strategy. Under such a strategy, once the insurer's reserve upcrosses the level b, the excess of the reserve over is paid off as dividends. We derive difference equations for both the expected discounted penalty function and the expected present value of dividend payments. Solution procedures for these difference equations are provided. When the joint distribution of the interclaim time and claim size is a finite mixture of bivariate geometric distributions, closed-form expressions are given. Numerical results for several sets of parameters are also provided to illustrate the applicability of the results obtained.

A generalization of Basu-Dhar’s bivariate geometric distribution to the trivariate case

In this paper, it is introduced a new parametric distribution to be used in multivariate lifetime data analysis as an alternative for the use of some existing multivariate parametric models as the popular multivariate normal distribution which is the most widely used model assumed in the analysis of continuous multivariate data. Although the normal multivariate distribution has univariate marginal normal probability distributions and simple interpretations for all their parameters, it may not be well fitted by many data sets, especially in survival data applications, usually considering logarithm transformed data. In many cases the use of parametric multivariate discrete models could be more appropriate for the data analysis. In this paper, it is introduced a generalization of the bivariate Basu-Dhar geometric distribution to a trivariate case applied to count data. Some properties of this trivariate geometric distribution, including its marginal probability distributions, order statistics distributions, the probability generating function and some simulation studies are presented. It is also presented some discussion on an extension of the trivariate case for the multivariate case. Classical and Bayesian inferences are presented assuming censored or uncensored observations. To illustrate the proposed methodology, two applications with real lifetime data are considered as examples.

Use of Basu-Dhar bivariate geometric distribution in the analysis of the reliability of component series systems

PurposeThe purpose of this paper is to provide a new method to estimate the reliability of series system by using a discrete bivariate distribution. This problem is of great interest in industrial and engineering applications.Design/methodology/approachThe authors considered the Basu–Dhar bivariate geometric distribution and a Bayesian approach with application to a simulated data set and an engineering data set.FindingsFrom the obtained results of this study, the authors observe that the discrete Basu–Dhar bivariate probability distribution could be a good alternative in the analysis of series system structures with accurate inference results for the reliability of the system under a Bayesian approach.Originality/valueSystem reliability studies usually assume independent lifetimes for the components (series, parallel or complex system structures) in the estimation of the reliability of the system. This assumption in general is not reasonable in many engineering applications, since it is possible that the presence of some dependence structure between the lifetimes of the components could affect the evaluation of the reliability of the system.

Weighted bivariate geometric distribution: Simulation and estimation

In this paper, we develop a new discrete bivariate distribution, i.e., the weighted bivariate geometric distribution whose marginals are univariate weighted geometric distributions and show that the proposed distribution is more flexible and applicable than the classical bivariate geometric distribution, which is in fact a special case of the proposed model, and the bivariate Freund distributions. We shall determine several important distributional and reliability characteristics of the distributions. Further, we shall develop classical and Bayesian inferences for the unknown parameters. Extensive Monte Carlo simulations and analysis of a real data set are conducted.

A bivariate geometric distribution allowing for positive or negative correlation

In this paper, we propose a new bivariate geometric model, derived by linking two univariate geometric distributions through a specific copula function, allowing for positive and negative correlations. Some properties of this joint distribution are presented and discussed, with particular reference to attainable correlations, conditional distributions, reliability concepts, and parameter estimation. A Monte Carlo simulation study empirically evaluates and compares the performance of the proposed estimators in terms of bias and standard error. Finally, in order to demonstrate its usefulness, the model is applied to a real data set.

Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach

ABSTRACTThe modeling and analysis of lifetime data in which the main endpoints are the times when an event of interest occurs is of great interest in medical studies. In these studies, it is common that two or more lifetimes associated with the same unit such as the times to deterioration levels or the times to reaction to a treatment in pairs of organs like lungs, kidneys, eyes or ears. In medical applications, it is also possible that a cure rate is present and needed to be modeled with lifetime data with long-term survivors. This paper presented a comparative study under a Bayesian approach among some existing continuous and discrete bivariate distributions such as the bivariate exponential distributions and the bivariate geometric distributions in presence of cure rate, censored data and covariates. In presence of lifetimes related to cured patients, it is assumed standard mixture cure rate models in the data analysis. The posterior summaries of interest are obtained using Markov Chain Monte Carlo methods. To illustrate the proposed methodology two real medical data sets are considered.

Basu-Dhar's bivariate geometric distribution in presence of censored data and covariates: some computational aspects

Some computational aspects to obtain classical and Bayesian inferences for the Basu and Dhar (1995) bivariate geometric distribution in presence of censored data and covariates are discussed in this paper. The posterior summaries of interest are obtained using standard existing MCMC (Markov Chain Monte Carlo) simulation methods available in popular free softwares as the OpenBugs software and the R software. Numerical illustrations are introduced considering simulated and real datasets showing that the use of discrete bivariate distributions may be a good alternative to the use of continuous bivariate distributions, in many areas of application.

Bivariate discrete generalized exponential distribution

ABSTRACTIn this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.

A Bayesian analysis for the bivariate geometric distribution in the presence of covariates and censored data

In this paper, we introduce a Bayesian analysis for bivariate geometric distributions applied to lifetime data in the presence of covariates and censored data using Markov Chain Monte Carlo (MCMC) methods. We show that the use of a discrete bivariate geometric distribution could bring us some computational advantages when compared to standard existing bivariate exponential lifetime distributions introduced in the literature assuming continuous lifetime data as for example, the exponential Block and Basu bivariate distribution. Posterior summaries of interest are obtained using the popular OpenBUGS software. A numerical illustration is introduced considering a medical data set related to the recurrence times of infection for kidney patients.

Analyzing dependence in incidence of diabetes and heart problem using generalized bivariate geometric models with covariates

ABSTRACTFor analyzing incidence data on diabetes and health problems, the bivariate geometric probability distribution is a natural choice but remained unexplored largely due to lack of models linking covariates with the probabilities of bivariate incidence of correlated outcomes. In this paper, bivariate geometric models are proposed for two correlated incidence outcomes. The extended generalized linear models are developed to take into account covariate dependence of the bivariate probabilities of correlated incidence outcomes for diabetes and heart diseases for the elderly population. The estimation and test procedures are illustrated using the Health and Retirement Study data. Two models are shown in this paper, one based on conditional-marginal approach and the other one based on the joint probability distribution with an association parameter. The joint model with association parameter appears to be a very good choice for analyzing the covariate dependence of the joint incidence of diabetes and heart diseases. Bootstrapping is performed to measure the accuracy of estimates and the results indicate very small bias.

Tests of Hypotheses for the Parameters of a Bivariate Geometric Distribution

Tests of Hypotheses for the Parameters of a Bivariate Geometric Distribution

Bivariate Conway–Maxwell–Poisson distribution: Formulation, properties, and inference

The bivariate Poisson distribution is a popular distribution for modeling bivariate count data. Its basic assumptions and marginal equi-dispersion, however, may prove limiting in some contexts. To allow for data dispersion, we develop here a bivariate Conway–Maxwell–Poisson (COM–Poisson) distribution that includes the bivariate Poisson, bivariate Bernoulli, and bivariate geometric distributions all as special cases. As a result, the bivariate COM–Poisson distribution serves as a flexible alternative and unifying framework for modeling bivariate count data, especially in the presence of data dispersion.