Discriminating between the extended exponential geometric distribution and the gamma distribution

Abstract

The proposed test called ELT is a goodness of fit test for discriminating between the family of the extended exponential geometric (EEG) distributions and the family of the Gamma distributions. The corresponding test statistic is developed based on the empirical Laplace transform () ktx e (, ) txwhere the argument value maximizes n X () ; t L t and is replaced by its MLE . It is verified that converges in distribution to (0 , 1). Performance of the proposed test is compared via Monte Carlo studies to that of the Kolmogorov – Smirnov (KS) test and the test based on the empirical moment generating function (EMGF) proposed by Epps et al. (1982) in two aspects, namely, estimated values of Type I error rates and power of the tests for various situations. It is found that the ELT and EMGF test can control the Type I error rate much better than the KS test in all situations. In addition, the ELT test seems to be more conservative than the EMGF test, that is, values of empirical Type I error rate ( () ) of ELT test are less than those of EMGF test in many situations especially when values of shape parameter are small ( = 0.1). However, the differences are small and close to zero as gets large. Further, under the alternative distribution of a Gamma family, empirical power values of the KS test are somewhat higher than those of the ELT and EMGF tests when sample size is small, but as increases, empirical power values of the ELT and EMGF tests increase and are much higher than those of the KS test. It is noticed that empirical power values of the ELT are slightly larger than the EMGF tests in many cases but the differences are not significant at 5% level.