Smooth tests of fit for more flexible alternatives to the exponential and Poisson: the Lindley and Poisson-Lindley distributions

We consider the little - known one parameter Lindley and Lindley - Poisson distributions. These distributions may be of interest as they appear to be more flexible than the exponential and Poisson distributions , the Lindley fitting more data than the exponential and the Lindley - Poisson fitting more data than the Poisson. We give smooth tests of fit for each of these distributions. The smooth test for the Lindley has power comparable with the Anderson - Darling test. Advantages of the smooth test are discussed. Examples that illustrate the flexibility of the two distributions are given.