Approximate MLEs for the location and scale parameters of the Poisson-half-logistic distribution

The application of compound distributions has recently increased due to the flexibility in fitting actual data in vari-ous fields such as economics, insurance, etc. Poisson-half-logistic distribution is one of these distributions with an increasing-constant hazard rate that can be used in parallel systems and complementary risk models. Because of the complexity of the form of this distribution, it is not possible to obtain classical parameter estimates (such as MLE) by the analytical method for the location and scale parameters. We present a simple way of deriving explicit estimators by approximating the likelihood equations appropriately. This paper presents the AMLE (Approximate Maximum Like-lihood Estimator) method to estimate the location and scale parameters. Using simulation, we show that this method is as efficient as the maximum likelihood estimators (MLEs). We obtain the variance of estimators from the inverse of the observed Fisher information matrix, and we see that when sample size increases, bias and variance of these estimators, and hence MSEs of parameters, decrease. Some pivotal quantities are proposed for finding confidence intervals for location and scale parameters based on asymptotic normality. From the coverage probability, the MLEs do not work well, especially for the small sample sizes; thus, simulated percentiles based on the Monte Carlo method are used to improve the coverage probability. Finally, we present a numerical example to illustrate the methods of inference developed here.