Bias-corrected maximum likelihood estimation of the parameters of the complex Bingham distribution

Abstract

In this paper, some bias correction methods are considered for parameter estimation of the complex Bingham distribution. The first method relies on the bias correction formula proposed by Cordeiro and Klein [Statistics & Probability Letters 19 (1994) 169–176]. The second method uses the formulas proposed by Kume and Wood [Statistics & Probability Letters 77 (2007) 832–837] for calculating the derivatives of the log likelihood function. The third method is based on the saddlepoint approximation proposed by Kume and Wood [Biometrika 92 (2005) 465–476]. Bootstrap bias correction methods due to Efron [The Annals of Statistics 7 (1979) 1–26] are also considered. Simulation experiments are used to compare the bias correction methods. In all cases, the analytical and bootstrap bias correction methods have smaller mean square errors. Since the dominant eigenvalue is used to obtain the mean shape, which has practical relevance, it is a key issue for comparing the estimators. The numerical results indicate that the bootstrap methods have a slightly better performance for the dominant eigenvalue.