Bivariate Poisson Distribution Research Articles

Implementation of a Parallel Algorithm to Simulate the Type I Error Probability

Simulating the probability of type I error is a powerful statistical tool that allows confirming if the statistical test achieves the established nominal level. However, its computational implementation has the drawback of significantly long execution times. Therefore, this article analyzes the performance of two parallel implementations (parRapply and boot) which significantly reduce the execution time of simulations of type I error probability for a goodness-of-fit test for the bivariate Poisson distribution. The results obtained demonstrate how the parallelization strategies accelerate the simulations, reducing the time by 50% to 90% when using 2 to 12 processors running in parallel. This reduction is graphically evidenced as the execution time of the analyzed parallel versions fits almost perfectly (R2≈0.999) to the power model y=apb, where p is the number of processors used, and a>0 and b<0 are the constants of the model. Furthermore, it is shown that the parallelization strategies used scale with an increasing number of processors. All algorithms were implemented in the R programming language, and their code is included at the end of this article.

Regression to the mean effect for bi-variate dispersed count data

Regression to the mean (RTM) occurs when extreme observations upon remeasurement are found closer to the population mean. Interventions are usually applied to subjects based on a cut-off point to check it effectiveness. The average difference of the pre-post observations called the total effect is the sum of RTM and intervention effects and needs to be accounted for RTM to avoid incorrect conclusions and unbiasedly estimate the intervention effect. The existing methods assume equidispersion and strict positive correlation for bivariate count data. This study considers a bivariate Poisson distribution which takes both the dispersion and negative correlation into consideration. Expressions for the total effect are derived and partitioned into RTM and intervention effects. The maximum likelihood estimators of the RTM and intervention effects are found numerically, and their asymptotic properties are verified via simulations. Data on the English premier league are used to estimate the RTM and intervention effects.

The Circumstance-Driven Bivariate Integer-Valued Autoregressive Model.

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule-Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated.

On a Bivariate Katz's Distribution Constructed by the Trivariate Reduction Method

In this paper, we propose the bivariate distribution of the Katz distribution [1] constructed by the trivariate reduction method, method developed in [6] and used in [5] to give an equivalent definition of the bivariate Poisson distribution [7, 8]. The constructed distribution includes, in particular, the bivariate Poisson distribution [5] and has interesting properties. For the estimation of the parameters, we used two methods: the method of moments and the maximum likelihood method using the EM algorithm. An application to concrete data has been made in order to carry out a comparative study between bivariate Poisson and Katz distributions, and we discuss the likelihood-ratio test, which assesses the goodness of fit of two competing statistical models.

On classical and Bayesian inference for bivariate Poisson conditionals distributions: theory, methods and applications

Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics, etc., to name but a few) and the bivariate Poisson distribution which is a generalization of the Poisson distribution plays an important role in modeling such data. In this article, we consider the inferential aspect of a bivariate Poisson conditionals distribution for which both the conditionals are Poisson but the marginals are typically non-Poisson. It has Poisson marginals only in the case of independence. It appears that a simple iterative procedure under the maximum likelihood method performs quite well as compared with other numerical subroutines, as one would expect in such a case where the MLEs are not available in closed form. In the Bayesian paradigm, both conjugate priors and non-conjugate priors have been utilized and a comparison study has been made via a simulation study. For illustrative purposes, a real-life data set is re-analyzed to exhibit the utility of the proposed two methods of estimation, one under the frequentist approach and the other under the Bayesian paradigm.

Multivariate control charts for monitoring a bivariate correlated count process with application to meningococcal disease.

In recent years, with the increasing number and complexity of infectious diseases, the idea of using control charts to monitor public health and disease has been proposed. In this paper, we study multivariate control charts for monitoring a bivariate integer-valued autocorrelation process with bivariate Poisson distribution and select the optimal control scheme by comparing the performance of control charts. Furthermore, the meningococcal patient event in two states in Australia serves as an example to illustrate the application of these methods. The results show that the D exponentially weighted moving average control scheme can detect the changes in the mean value faster, which is a significant advantage.

THE BIVARIATE EXTENDED POISSON DISTRIBUTION OF TYPE 2

In this paper we undertake the construction of a bivariate distribution generalising the univariate extended Poisson distribution by using the method of crossing laws, a method highlighted in [7]. We will call this law "the bivariate extended Poisson distribution of type 2", in reference to "the bivariate extended Poisson law of type 1" highlighted in [4]. We have shown that this law is a member of the family of bivariate Poisson distributions. Functional relations will be established between the two distributions.

A Poisson cokriging method for bivariate count data

Bivariate spatially correlated count data appear naturally in several domains such as ecology, economy and epidemiology. Current methods for analysing such data lack simplicity, interpretability and computational awareness. This paper introduces Poisson cokriging, a bivariate geostatistical technique to model and predict spatially correlated count variables. Our method exploits classical geostatistical theory and the bivariate Poisson distribution to propose an adaptation of cokriging when the underlying process follows a bivariate Poisson structure. A simulation study and a real data application using counts from two mosquito-borne diseases in Colombia showed that our model successfully performs spatial predictions at unobserved locations under different settings. We demonstrate the competitive convenience of Poisson cokriging in filtering rates and modelling highly variant population sizes against traditional geostatistical methods. We conclude that Poisson cokriging improves prediction accuracy and reduces variance prediction errors in comparison with ordinary cokriging.

Effusion of stochastic processes on a line

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.

ON A BIVARIATE KATZ’S DISTRIBUTION

In this paper, we propose the bivariate distribution of the univariate Katz distribution [7] using the technique of the product of marginal distributions by a multiplicative factor. This method has been examined in [11] and used in [9] to construct a bivariate Poisson distribution. The obtained model is a good way to unify bivariate Poisson, bivariate binomial and bivariate negative binomial distributions and has interesting properties. Among others, the correlation coefficient of the obtained model can be either positive, negative, or null, and the necessary condition of zero correlation is a necessary and sufficient condition for independence. We used two methods to estimate the parameters: the method of moments and the maximum likelihood method. An application to concrete insurance data has been made. This data concerns natural events insurance in the USA and third-party liability automobile insurance in France [13].

Evaluating capability of a bivariate zero-inflated poisson process

A zero-inflated Poisson (ZIP) distribution is commonly used for modelling zero-inflated process data with single type of defect, and for developing appropriate tools for instituting statistical process control of manufacturing processes. However, in reality, such manufacturing scenarios are very common where more than one type of defect can occur. For example, occurrences of defects like solder short circuits (shorts) and absence of solder (skips) are very common on printed circuit boards. In literature, different forms of bivariate zero-inflated Poisson (BZIP) distributions are proposed, which can be used for modelling the manufacturing scenarios where two types of defects can occur. Control charts are designed for monitoring for such processes using BZIP models. Although evaluation of capability is an integral part of statistical process control of a manufacturing process, researchers have given very little effort on this aspect of zero-inflated processes. Only a few articles attempted to evaluate the capability of a univariate zero-inflated process and no work is reported on evbaluating capability of a bivariate zero-inflated process. In this paper, a methodology for measuring capability of a bivariate zero-inflated process is presented. The proposed methodology is illustrated using two case studies.

Construction and Selection of Double Inspection Single Sampling Plan for an Independent Process Using Bivariate Poisson Distribution

There are more sampling concepts active in production Industries, for inspecting the samples and analysing performance of the population. Also the sampling plans reduce errors in the production and produce the error free products. In this study, construction and selection of Double Inspection with reference to Single Sampling Plan i.e., DISSP, by attribute are investigated by using the Bivariate Poisson distribution. The Methodology, DISSP, was proposed based on two quality characteristics of the same sample size, and the planning parameters (n, C₁, C₂) are based on the operating characteristics, the conventional two-point condition by the planning table parameters (AQL and LQL). It is based on selected quality requirements and risks designed to allow manufacturers to easily determine the required sample size and corresponding acceptance criteria. A Comparison was done based on the efficiency of the plan with an existing single sampling plan and gave a numerical example to expose the operating tables. Also, the study shows the advantages of the proposed plan, and performance of the curves like, Operating characteristics, Average Outgoing Quality, and Average Total Inspection to expose the proposed double inspection sampling plan.

Bivariate Discrete Poisson-Lindley Distributions.

Two families of bivariate discrete Poisson–Lindley distributions are introduced. The first is derived by mixing the common parameter in a bivariate Poisson distribution by different models of univariate continuous Lindley distributions. The second is obtained by generalizing a bivariate binomial distribution with respect to its exponent when it follows any of five different univariate discrete Poisson–Lindley distributions with one or two parameters. The use of probability-generating functions is mainly employed to derive some general properties for both families and specific characteristics for each one of their members. We obtain expressions for probabilities, moments, conditional distributions, regression functions, as well as characterizations for certain bivariate models and their marginals. An attractive property of all bivariate individual models is that they contain only two or three parameters, and one of them is readily estimated by simple ratios of their sample means. This feature, and since all marginal distributions are over-dispersed, strongly suggests their potential use to describe bivariate dependent count data in many different areas.

The Bivariate Extended Poisson Distribution of Type 1

In this paper, we will construct the bivariate extended Poisson distribution whichgeneralizes the univariate extended Poisson distribution. This law will be obtained by the method of the product of its marginal laws by a factor. This method was demonstrated in [7]. Thus we call the bivariate extended Poisson distribution of type 1 the bivariate extended Poisson distribution obtained by the method of the product of its marginal distributions by a factor. We will show that this distribution belongs to the family of bivariate Poisson distributions and and will highlight the conditions relating to the independence of the marginal variables. A simulation study was realised.

A flexible bivariate distribution for count data expressing data dispersion

The bivariate Poisson distribution is a natural choice for modeling bivariate count data. Its constraining assumption, however, limits model flexibility in some contexts. This work considers the trivariate reduction method to construct a Bivariate Conway-Maxwell-Poisson (BCMP) distribution, which accommodates over- and under-dispersed data. The approach produces marginals that have a flexible form which includes several special case distributions for certain parameters. Moreover, this BCMP model performs well relative to other bivariate models for count data, including BCMP models based on different methods of construction. As a result, the trivariate-reduced BCMP distribution is a flexible alternative for modeling bivariate count data containing data dispersion.

A Note on the (Weighted) Bivariate Poisson Distribution

In the recent statistical literature, the univariate Poisson distribution has been generalized by many authors, among them: the univariate weighted Poisson distribution [13], the generalized univariate Poisson distribution [7], the bivariate Poisson distribution according to Holgate [11], the bivariate Poisson distribution according to Lakshminarayana, Pandit and Srinivasa Rao [15], the bivariate Poisson distribution according to Berkhout and Plug [4], the bivariate weighted Poisson distribution according to Elion et al. [8] and the generalized bivariate Poisson distribution according to Famoye [9]. In this paper, We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distributions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout and Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N.

Comparison of BINAR(1) models with bivariate negative binomial innovations and explanatory variables

The bivariate integer-valued autoregressive model of order 1 (BINAR(1)) is popular in fitting bivariate time series of counts, and the bivariate negative binomial (BNB) distribution can be chosen as its innovation's distribution, which is more flexible than the traditional bivariate Poisson distribution. It is well known that BNB distributions can be constructed in different ways, and these distributions will be reviewed in this paper. Performances of BINAR(1) models based on these BNB distributions with explanatory variables being included in the survival probability are compared. To estimate unknown parameters, the conditional maximum likelihood method is considered and evaluated by Monte Carlo simulations. Two sales counts are used to compare performances of the above models, and some interesting conclusions are also given.

The moving particles method for reliability estimation: recent advances

AbstractIn the following, recent advances of the moving particles method are highlighted. A multi level version of the estimator is introduced that balances the statistical error and the numerical approximation error by computing a telescoping sum of estimates for the number of moves. An extension to general model classes in the sense of a multi fidelity method is obtained based on the estimation of bivariate Poisson distributions for censored data. Finally, the local sensitivity of the reliability estimate with respect to the parameters of the probability characteristics of the model input is considered. It is shown how to obtain estimates of the local sensitivity without any additional function evaluations.

The New Bivariate Conway-Maxwell-Poisson Distribution Obtained by the Crossing Method

Kimberly et al. had proposed in 2016 a bivariate function as a bivariate Conway-Maxwell-Poisson distribution (COM-Poisson) using the generalized bivariate Poisson distribution and the probability generating functions of the follow distributions: bivariate bernoulli, bivariate Poisson, bivariate geometric and bivariate binomial. By examining this paper we have shown that this bivariate function is constant and it double series is divergent, when it should have been 1. To overcome this deadlock, we propose a new bivariate Conway-Maxwell-Poisson distribution which is definetely a probability distribution based on the crossing method, method highlighted by Elion et al. in 2016 and revisited by Batsindila et al. and Mandangui et al. in 2019. And this is the purpose of this paper. To this bivariate distribution is attached two generalized linear models (GLM) whose resolution allows us to highlight, not only the independence between the variables forming the couple, but also the effect of the factors (or predictors) on these variables. The resulting correlation is negative, zero or positive depending on the values of a parameter; in particular for the bivariate Poisson distribution according to Berkhout and Plug. A simulation of data will be given at the end of the article to illustrate the model.

A new bivariate Poisson distribution via conditional specification: properties and applications

ABSTRACT In this article, we discuss a bivariate Poisson distribution whose conditionals are univariate Poisson distributions and the marginals are not Poisson which exhibits negative correlation. Some useful structural properties of this distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of negative correlation, marginal over-dispersion, distribution of sum and 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. Parameter estimation with maximum likelihood is implemented. Copula-based simulation experiments are carried out using Bivariate Normal and the Farlie–Gumbel–Morgenstern copulas to assess how the model behaves in dealing with the situation. Finally, the distribution is fitted to seven bivariate count data sets with an inherent negative correlation to illustrate suitability.