Improved maximum-likelihood estimation of the shape parameter in the Nakagami distribution

Abstract

We develop and evaluate analytic and bootstrap bias-corrected maximum-likelihood estimators for the shape parameter in the Nakagami distribution. This distribution is widely used in a variety of disciplines, and the corresponding estimator of its scale parameter is trivially unbiased. We find that both ‘corrective’ and ‘preventive’ analytic approaches to eliminating the bias, to O(n ⁻²), are equally, and extremely, effective and simple to implement. As a bonus, the sizeable reduction in bias comes with a small reduction in the mean-squared error. Overall, we prefer analytic bias corrections in the case of this estimator. This preference is based on the relative computational costs and the magnitudes of the bias reductions that can be achieved in each case. Our results are illustrated with two real-data applications, including the one which provides the first application of the Nakagami distribution to data for ocean wave heights.

Keywords:

• maximum likelihood
• Monte Carlo simulation
• bootstrap

Acknowledgements

We are grateful to Lief Bluck and Kees van Kooten for arranging the computing facilities needed to undertake this research in a timely manner. We also thank Professor Hisashi Nakahara and Professor P. Mohana Shankar for helpful correspondence.

Notes