On the MLE of the Waring distribution

The two-parameter Waring is an important heavy-tailed discrete distribution, which extends the famous Yule-Simon distribution and provides more flexibility when modelling the data. The commonly used EFF (Expectation-First Frequency) for parameter estimation can only be applied when the first moment exists, and it only uses the information of the expectation and the first frequency, which is not as efficient as the maximum likelihood estimator (MLE). However, the MLE may not exist for some sample data. We apply the profile method to the log-likelihood function and derive the necessary and sufficient conditions for the existence of the MLE of the Waring parameters. We use extensive simulation studies to compare the MLE and EFF methods, and the goodness-of-fit comparison with the Yule-Simon distribution. We also apply the Waring distribution to fit an insurance data.