On the MLE of the Waring distribution

Abstract

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.

Keywords:

• Maximum likelihood estimator
• heavy-tailed discrete distribution
• Waring distribution

Acknowledgements

The authors would like to thank two anonymous reviewers, an associate editor and the editor for constructive comments and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by National Natural Science Foundation of China [Grant Numbers 11671096, 11690013, 11731011, 11871376] and Natural Science Foundation of Shanghai [Grant Number 21ZR1420700].