Tests of exponentiality against some parametric over/under-dispersed life time models

We develop tests of goodness of fit of the exponential model against some over/under dispersion family of distributions. In particular, we develop 3 score test statistics and 3 likelihood ratio statistics. These are (S-1, L-1), (S-2, L-2), and (S-3, L-3) based on a general over-dispersed family of distributions, two specific over/under dispersed exponential models, namely, the gamma and the Weibull distributions, respectively. A simulation study shows that the statistics S-3 and L-3 have best overall performance, in terms of both, level and power. However, the statistic L3 can be liberal in some instances and it needs the maximum likelihood estimates of the parameters of the Weibull distribution as opposed to the statistic S3 which is very simple to use. So, our recommendation is to use the statistic S3 to test the fit of an exponential distribution over any over/under-dispersed exponential distribution.