Bivariate Zero-Inflated Power Series Distribution

Many researchers have discussed zero-inflated univariate distributions. These univariate models are not suitable, for modeling events that involve different types of counts or defects. To model several types of defects, multivariate Poisson model is one of the appropriate models. This can further be modified to incorporate inflation at zero and we can have multivariate zero-inflated Poisson distribution. In the present article, we introduce a new Bivariate Zero Inflated Power Series Distribution and discuss inference related to the parameters involved in the model. We also discuss the inference related to Bivariate Zero Inflated Poisson Distribution. The model has been applied to a real life data. Extension to k-variate zero inflated power series distribution is also discussed.


Introduction
In a manufacturing process there may exist several types of (say m) defects-for example, solder short circuits, solder voids, absence of solder etc. on one printed circuit board.These defects cause different types of product failure and generate different types of equipment problems.In the above example there can be only one type of defect which occurs more frequently and the other defects occurs very rarely.Another situation could be both types of defects occur rarely and so on.To model several types of defects, multivariate Poisson model is one of the appropriate models to use.This can further be modified to incorporate inflation at zero and we can have multivariate zero-inflated Poisson (MZIP) distribution.There are several ways to construct MZIP distributions.In the literature, Chin-Shang et al. [1] have discussed various types of MZIP models and investigated their distributional properties.Deshmukh and Kasture [2] have studied bivariate distribution with truncated Poisson marginal distributions.Gupta et al. [3] have considered inflated distributions at the point zero and studied the structural properties of the inflated distribution.Gupta et al. [4] have discussed score test for zero-inflated generalized Poisson regression model.Holgate [5] described the es-timation of covariance parameter of bivariate Poisson distribution by iterative method.Lambert [6] considered zero-inflated Poisson regression model.Laxminarayana et al. [7] have studied bivariate Poisson distribution and the distributional properties of the model.Patil and Shirke [8] studied testing parameter of the power series distribution of a zero-inflated power series model.Patil and Shirke [9] also studied equality of inflation parameters of two zero-inflated power series distributions.It appears that majority of the study in the literature is restricted to Poisson distribution and its extension to multivariate set up.Relatively less has been reported for the family of distributions containing other distributions.
In this article, we introduce a new Bivariate Zero-Inflated Power Series Distribution (BZIPSD) and discuss inference related to the parameters involved in the same.The rest of the paper is organized as follows.Section 2, introduces the BZIPSD along with moments of the same.Section 3, deals with inference related to the parameters involved in the BZIPSD.In Section 4, we discuss inference related to Bivariate Zero-Inflated Poisson Distribution (BZIPD).The data set reported by Arbous and Kerrich [10] is modeled by Bivariate Zero Inflated Poisson Distribution.The paper concludes with generalization to multivariate setup.

Moment Generating Function
of (X,Y) is sults on the remaining two can be obtained analogously.
The moment generating function Therefore, we have ,0 and the correlation coefficient is (2.7)

  
The likelihood funct e is given by.ion for the observed random sampl where if 0 and otherwise.The co din unct ven by, give the following equations.

Bivariate Zero-Inflated Poisson Distribution
Let us set     in the model (2.1).Then we get BZIPD with probability mass function.
, 0,0 where 1 1 x y c e The moment generating function of (X,Y) is

t x t y
x y e P x e P y It is clear from the expressions of moment generating functions of X and that the marginal distributions of X and Y are univariate zero-inflated power series distributions with param ters The correlation coefficient is turns out to be Remark 1: When there is no inflation , the cor-  , , ; , e e e e e x y where x y  y ot 0 and otherwise.The corresponding log likelihood is given by, The mles of the parameters can be obtained     From the above equations, it is clear that Equations (4.6) to (4.9) are non-linear in nature.Solving these equations is computationally cumbersome.Laxminarayan et al. [7], adopt method of moments for the model without inflation parameter (i.e. ).In their model they have used estimates based on Method of Moment Estimators (MME), which coincide with Maximum Likelihood Estimators (MLE) of the marginal distributions.This is not the case for the joint distribution.We have to solve four equations simultaneously in order to et the MLEs.In the following we obtain maximum likelihood estimators for the following example and test for goodness of fit.

An Application
The data set in Table 1 reported by Arbous and Kerrich [10], .Using these mles we fit the data of the marginal distribution of X to ZIPD , we get Chi square statistic = 0.74843 and P value = 0.3869.If we fit the data by 122 railway men in consecutive periods of 6 and 5 years.X is the accident distribution of 122 railway men during 1937-1942 and Y is the accident distribution of 122 railway men during 1943-1947.By assuming marginal distributions of X is ZIPD the marginal distribution of Y to ZIPD we get Chi square statistic = 0.6065 and P value = 0.4360.

Table 2 . Expected frequencies using BZIPD.
Inflated Negative Binomial Distribution or k-variate zero inflated Poisson distribution can also be