On maximum likelihood estimation of the general projected normal distribution

Abstract

The general projected normal distribution is a flexible distribution family that is widely used for modelling circular data. It is constructed by projecting the bivariate normal distribution onto the unit circle. Maximum likelihood estimation for the general projected normal distribution was considered for the special cases of $\mathbf{\Sigma }=\mathbf{\text{I}}$ as well as$\mathbf{\Sigma }={\sigma }^2\mathbf{\text{I}}$. In this study, we consider the problem of maximum likelihood estimation assuming general form of $\mathbf{\Sigma }$. EM-algorithm and Newton Raphson method are formulated and used to obtain approximations for the final estimators. A simulation study is conducted to evaluate the performance of the estimated parameters. In general, the obtained estimators show consistent behaviour where EM-algorithm has better performance compared to the Newton Raphson method. Moreover, extension for the regression problem is introduced where EM-algorithm is developed. Two examples are provided to demonstrate the proposed estimators in real applications.

KEYWORDS:

• Circular data
• angular mean direction
• mean resultant length
• EM-algorithm
• Newton Raphson method

Acknowledgment

The authors thank the anonymous reviewer for his/her comments and suggestions which enhanced the value and clarity of the paper.

Disclosure statement

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