Asymptotic tests for poisson distribution

Abstract

Poisson distribution is well used as a standard model for analyzing the count data in many fields such as environmental sciences, epidemiology, manufacturing applications, econometrics and finance. The importance for testing the fit of the observations arises from a Poisson distribution. In general, the tests related to the Index of Dispersion (ID) concerning the mean and variance is quite powerful against a wide variety of alternatives except when the ID is closer to 1 (Gurtler and ɺɺ Henze, 2000: 219; Karlis and Xekalaki, 2000: 381; Meintanis and Nikitin, 2008: 3729). However, there are many discrete distributions, which have a property of equal or almost equal dispersion (ID is close to 1). As a consequence, the tests related to the ID are not able to discriminate between the Poisson distribution and some other discrete distributions. In this dissertation, the two proposed alternative tests of fit for Poisson distribution are based on the characterization of skewness and the coefficient of variation, called SK and CSK, and are asymptotically normally distributed. The asymptotic level alpha of these statistics desired values of nominal levels. In theory, the comparative the powers of the proposed tests, in respect to other tests are too complicated and not worth to being considered because of their complication. Therefore, in practice, the asymptotic powers are investigated against some fixed alternatives in terms of the Absolute Inverse Coefficient of Variation, Abs(Inv_CV). The Abs(Inv_CV) is greater than the critical value of the test, which indicates that the power tends to 1. For discrete uniform alternative, the Abs(Inv_CV) of the procedure test of the CSK is greater than the SK except in family of under-dispersion distributions. Simulation studies suggest that the empirical Type I error rates of the proposed tests are expected at the 5% significance level when the sample of sizes are 50 and 100. Results of the empirical power studies show that a sample of size 100 is sufficient for tests involving the SK and CSK tests. The proposed tests are sensitive to mixed-dispersion alternatives for discrete uniform, beta-binomial and zero-inflated generalized Poisson distributions. In some situations the CSK test offers a more powerful test than the SK test and in other cases the power of these two tests are not appreciably different. The tests related to the ID are quite powerful against a wide variety of alternatives when the ID is further away from 1. However, the proposed tests do not perform as well for under-dispersion and over-dispersion alternatives, and seem to be nearly equivalent to the tests related to the ID when the range is 0.95 to 1.05. Then, the CSK test statistic is very practical, to test the observation departures from Poissonity when the alternative is unknown and the ID is closer to 1.