×

Intrinsic covariance matrix estimation for multivariate elliptical distributions. (English) Zbl 1439.62153

Summary: The property of statistical models not depending on the coordinate systems or model parametrization is one main interest of intrinsic inference in statistics. The intrinsic covariance matrix estimation is addressed for multivariate elliptical distributions in this paper. An optimal intrinsic covariance estimator is derived in the sense of minimizing the mean square Rao distance, and proved to own intrinsic unbiasedness. Specifically, the intrinsically unbiased estimators for elliptical distributions and mixture elliptical distributions are developed.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H12 Estimation in multivariate analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amari, S., Differential-Geometrical Methods in Statistics (1990), Springer: Springer Berlin, Germany · Zbl 0701.62008
[2] Atkinson, C.; Mitchell, A. F.S., Rao’s distance, Sankhyā, 43, 3, 345-365 (1981) · Zbl 0534.62012
[3] Berkane, M.; Oden, K.; Bentler, P. M., Geodesic estimation in elliptical distributions, J. Multivariate Anal., 63, 1, 35-46 (1997) · Zbl 0897.62052
[4] Bernardo, J. M., Intrinsic point estimation of the normal variance, (Upadhyay, S. K.; Singh, U.; Dey, D. K., Bayesian Statistics and Its Applications (2007), Anamaya Publishers: Anamaya Publishers New Delhi, India), 110-121
[5] Bernardo, J. M., Objective Bayesian point and region estimation in location-scale models, SORT: Stat. Oper. Res. Trans., 31, 1, 3-44 (2007) · Zbl 1274.62195
[6] Bernardo, J. M., Integrated objective Bayesian estimation and hypothesis testing, (Bernardo, J. M.; Bayarri, M. J.; Berger, J. O.; Dawid, A. P.; Heckerman, D.; Smith, A. F.M.; West, M., Bayesian Statistics 9: Proceedings of the Ninth Valencia International Meeting (2011), Oxford University Press: Oxford University Press New York, NY, USA), 1-68
[7] Bernardo, J. M.; Juárez, M. A., Intrinsic estimation, (Bernardo, J. M.; Bayarri, M. J.; Berger, J. O.; Dawid, A. P.; Heckerman, D.; Smith, A. F.M.; West, M., Bayesian Statistics 7: Proceedings of the Seventh Valencia International Meeting (2003), Oxford University Press: Oxford University Press New York, NY, USA), 465-476
[8] Cheng, Y.; Wang, X.; Morelande, M.; Moran, B., Information geometry of target tracking sensor networks, Inf. Fusion, 14, 3, 311-326 (2013)
[9] Fujikoshi, Y.; Ulyanov, V. V.; Shimizu, R., Multivariate Statistics: High-Dimensional and Large-Sample Approximations (2010), John Wiley & Sons: John Wiley & Sons Hoboken, NJ, USA · Zbl 1304.62016
[10] García, G.; Oller, J. M., What does intrinsic mean in statistical estimation?, SORT: Stat. Oper. Res. Trans., 30, 2, 125-146 (2007) · Zbl 1274.62167
[11] Ilea, I.; Bombrun, L.; Terebes, R.; Borda, M.; Germain, C., An M-estimator for robust centroid estimation on the manifold of covariance matrices, IEEE Signal Process. Lett., 23, 9, 1255-1259 (2016)
[12] Moakher, M., A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 26, 3, 735-747 (2005) · Zbl 1079.47021
[13] Oller, J. M., On an intrinsic analysis of statistical estimation, (Cuadras, C.; Rao, C., Multivariate Analysis: Future Directions 2 (1993), North-Holland: North-Holland Amsterdam, Holland), 421-437 · Zbl 0810.62031
[14] Oller, J. M.; Corcuera, J. M., Intrinsic analysis of statistical estimation, Ann. Statist., 23, 5, 1562-1581 (1995) · Zbl 0843.62027
[15] Protter, M. H.; Morrey, C. B., Intermediate Calculus (1985), Springer: Springer New York, NY, USA · Zbl 0555.26002
[16] Rao, C. R., Information and the accuracy attainable in the estimation of statistical parameters, News Bull. Calcutta Math. Soc., 37, 81-91 (1945) · Zbl 0063.06420
[17] Robert, C. P., Intrinsic losses, Theory and Decision, 40, 2, 191-214 (1996) · Zbl 0848.90010
[18] Rong, Y.; Tang, M.; Zhou, J., Intrinsic losses based on information geometry and their applications, Entropy, 19, 8, 405 (2017)
[19] Smith, S. T., Covariance, subspace, and intrinsic Cramér-Rao bounds, IEEE Trans. Signal Process., 53, 5, 1610-1630 (2005) · Zbl 1370.94242
[20] Sutradhar, B. C.; Ali, M. M., A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate \(t\) model, J. Multivariate Anal., 29, 1, 155-162 (1989) · Zbl 0667.62036
[21] Tang, M.; Rong, Y.; Zhou, J.; Li, X. R., Information geometric approach to multisensor estimation fusion, IEEE Trans. Signal Process., 67, 2, 279-292 (2019) · Zbl 1415.94250
[22] Tiao, G. G.; Cuttman, I., The inverted Dirichlet distribution with applications, J. Amer. Statist. Assoc., 60, 311, 793-805 (1965) · Zbl 0133.42504
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.