Quantifying the ratio-plot for the geometric distribution

Abstract

The geometric distribution is one of the most widely used count distributions. Novel goodness of fit tests for this distribution are suggested taking advantage of a characterization of that distribution in terms of a differential equation involving its probability generating function. Several ways of looking at the characterization allow us to derive six test statistics. The connection between some of these test statistics and the ratio-plot device is stated. The asymptotic null distributions of these test statistics are derived. However, they depend on the unknown parameter of the geometric law. A suitable parametric bootstrap is used to estimate consistently each null distribution. Moreover, the almost sure limits of the test statistics under alternatives are obtained. The finite sample performance of the bootstrap approximation is assessed via simulation. The powers of the new tests are numerically compared with that of some existing ones, exhibiting competitive behaviour. Some real-life data set applications are included.