Tests of fit for discrete distributions based on the probability generating function

Abstract

The use of the probability generating function in testing the fit of discrete distributions was proposed by Kocherlakota & Kocherlakota (1986), and further studied by Márques and Perez-Abreu (1989). In Rueda et al. (1991), a quadratic statistic to test the fit of a discrete distribution was proposed using the probability generating function and its empirical counterpart. This was illustrated for the Poisson case with known parameter. Here, we deal with some extensions: the Poisson case with unknown parameter and the negative Binomial distribution with known or unknown parameter p. We find the asymptotic distribution of the test statistic in each case, and show with the aid of some Monte Carlo studies the closeness of these asymptotic distributions.

A connection is established between this quadratic test and the Cramér von Mises test of fit described in Spinelli (1994) and Spinelli and Stephens (1997), thus providing additional insight into these procedures. Also, a correction is made on the expression of the covariance function of the empirical process as appeared in Rueda et al (1991). Finally, power comparisons are provided for the case of the Poisson test and some examples are given.

Keywords:

• Goodness of fit
• discrete distribution
• probability generating function
• empirical process