Goodness-of-fit tests for the bivariate Poisson distribution

Abstract

The bivariate Poisson distribution is commonly used to model bivariate count data. In this paper we study a goodness-of-fit test for this distribution. We also provide a review of the existing tests for the bivariate Poisson distribution, and its multivariate extension.The proposed test is consistent against any fixed alternative. It is also able to detect local alternatives converging to the null at the rate n − 1 2 . The bootstrap can be employed to consistently estimate the null distribution of the test statistic. Through a simulation study we investigated the goodness of the bootstrap approximation and the power for finite sample sizes.

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

“…Additionally, implementing the algorithm developed in this research in the C programming language is planned. Furthermore, developing a package in the R language that incorporates the goodness-of-fit test studied and others proposed in [1, [19][20][21] is also planned. y_max = max(X[2,]) frec = matrix(0,x_max-x_min+1,y_max-y_min+1) for(i in x_min:x_max){ p = i -x_min + 1 for(j in y_min:y_max){ q = j -y_min + 1 for (k in 1 for(j in y_min:y_max){ q = j -y_min + 1 if(frec[p,q]>0){ prob_ij = probPB(i,j,t1,t2,t3) prob_im1jm1 = probPB(i-1,j-1,t1,t2,t3) prob_im2jm2 = probPB(i-2,j-2,t1,t2,t3) prob_im2jm1 = probPB(i-2,j-1,t1,t2,t3) prob_im1jm2 = probPB(i-1,j-2,t1,t2,t3) prob_im1j = probPB(i-1,j,t1,t2,t3) prob_ijm1 = probPB(i,j-1,t1,t2,t3) rm = rm + frec[p,q]*prob_im1jm1/prob_ij sum1 = (prob_im2jm2 -prob_im2jm1 -prob_im1jm2)/prob_ij sum2 = prob_im1jm1*(prob_im1j + prob_ijm1 -prob_im1jm1)/(prob_ij^2) der = der + frec[p,q]*(sum1 + sum2)…”

Goodness-of-Fit Test for the Bivariate Hermite Distribution

“…According to Novoa-Muñoz in [13], the probability generating function (pgf) characterizes the distribution of a random vector and can be estimated consistently by the empirical probability generating function (epgf); the proposed test is a function of the epgf. This statistical test compares the epgf of the data with an estimator of the pgf of the BHD.…”

Goodness-of-fit Test for the Bivariate Hermite Distribution

“…According to Novoa-Muñoz in [12], the probability generating function (pgf) characterizes the distribution of a random vector and can be estimated consistently by the empirical probability generating function (epgf), the proposed test is a function of the epgf. This statistical test compares the epgf of the data with an estimator of the pgf of the BHD.…”