Bias-corrected maximum likelihood estimation of the parameters of the complex Bingham distribution. (English) Zbl 1381.62049

Summary: 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 G. M. Cordeiro and R. Klein [Stat. Probab. Lett. 19, No. 3, 169–176 (1994; Zbl 0791.62087)]. The second method uses the formulas proposed by A. Kume and the last author [ibid. 77, No. 8, 832–837 (2007; Zbl 1373.62239)] for calculating the derivatives of the log likelihood function. The third method is based on the saddlepoint approximation proposed by A. Kume and the last author [Biometrika 92, No. 2, 465–476 (2005; Zbl 1094.62063)]. Bootstrap bias correction methods due to B. Efron are also considered [Ann. Stat. 7, 1–26 (1979; Zbl 0406.62024)]. 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.


62F10 Point estimation
60E05 Probability distributions: general theory
62H11 Directional data; spatial statistics
Full Text: DOI Euclid


[1] Amaral, G. J. A., Florez, O. P. R. and Cysneiros, F. J. A. (2013). Graphical and numerical methods for detecting influential observations in complex Bingham data. Communications in Statistics. Simulation and Computation 42 , 1801-1814. · Zbl 1301.62052 · doi:10.1080/03610918.2012.677921
[2] Bartlett, M. S. (1953). Approximate confidence intervals. Biometrika 40 , 12-19. · Zbl 0050.36302 · doi:10.1093/biomet/40.1-2.12
[3] Bingham, C. (1974). An antipodally symmetric distribution on the sphere. The Annals of Statistics 2 , 1201-1225. · Zbl 0297.62010 · doi:10.1214/aos/1176342874
[4] Cordeiro, G. M. and Cribari-Neto, F. (2014). An Introduction to Bartlett Correction and Bias Reduction . Heidelberg: Springer. · Zbl 1306.62025 · doi:10.1007/978-3-642-55255-7
[5] Cordeiro, G. M. and Klein, R. (1994). Bias correction in ARMA models. Statistics & Probability Letters 19 , 169-176. · Zbl 0791.62087 · doi:10.1016/0167-7152(94)90100-7
[6] Cox, D. R. and Snell, E. J. (1968). A general definition of residuals (with discussion). Journal of the Royal Statistical Society, Series B, Statistical Methodology 30 , 248-275. · Zbl 0164.48903
[7] Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis . Chichester: John Wiley and Sons. · Zbl 0901.62072
[8] Efron, B. (1979). Bootstrap methods: Another look at the Jackknife. The Annals of Statistics 7 , 1-26. · Zbl 0406.62024 · doi:10.1214/aos/1176344552
[9] Kent, J. T. (1994). The complex Bingham distribution and shape analysis. Journal of the Royal Statistical Society, Series B, Statistical Methodology 56 , 285-299. · Zbl 0806.62040
[10] Kent, J. T., Constable, P. D. L. and Er, F. (2004). Simulation of the complex Bingham distribution. Statistics and Computing 14 , 53-57.
[11] Kume, A. and Wood, A. T. A. (2005). Saddlepoint approximations for the Bingham and Fisher-Bingham normalising constants. Biometrika 92 , 465-476. · Zbl 1094.62063 · doi:10.1093/biomet/92.2.465
[12] Kume, A. and Wood, A. T. A. (2007). On the derivatives of the normalising constant of the Bingham distribution. Statistics & Probability Letters 77 , 832-837. · Zbl 1373.62239
[13] Mardia, K. I. and Jupp, P. E. (2000). Directional Statistics . Chichester: Wiley. · Zbl 0935.62065
[14] Small, C. G. (1996). The Statistical Theory of Shape . New York: Springer. · Zbl 0859.62087
[15] Watson, G. S. (1965). Equatorial distributions on a sphere. Biometrika 52 , 193-201.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.