Testing Exponentiality Versus Pareto Distribution via Likelihood Ratio

Abstract

We consider the problem of maximum likelihood estimation of the parameters of the Pareto Type II (Lomax) distribution. We show that in certain parametrization and after modification of the parameter space to include exponential distribution as a special case, the MLEs of parameters always exist. Moreover, the MLEs have a non standard asymptotic distribution in the exponential case due to the lack of regularity. Further, we develop a likelihood ratio test for exponentiality versus Pareto II distribution. We emphasize that this problem is non standard, and the limiting null distribution of the deviance statistic in not chi-square. We derive relevant asymptotic theory as well as a convenient computational formula for the critical values for the test. An empirical power study and power comparisons with other tests are also provided. A problem from climatology involving precipitation data from hundreds of meteorological stations across North America provides a motivation for and an illustration of the new test.