Stein–Haff identity for the exponential family
Author:
G. Alfelt
Journal:
Theor. Probability and Math. Statist. 99 (2019), 5-17
MSC (2010):
Primary 62H12; Secondary 62C99
DOI:
https://doi.org/10.1090/tpms/1076
Published electronically:
February 27, 2020
MathSciNet review:
3908652
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper, the Stein–Haff identity is established for positive-definite and symmetric random matrices belonging to the exponential family. The identity is then applied to the matrix-variate gamma distribution, and an estimator that dominates the maximum likelihood estimator in terms of Stein’s loss is obtained. Finally, a simulation study is conducted in order to support the theoretical results.
References
- Taras Bodnar and Arjun K. Gupta, An identity for multivariate elliptically contoured matrix distribution, Statist. Probab. Lett. 79 (2009), no. 10, 1327–1330. MR 2522604, DOI https://doi.org/10.1016/j.spl.2009.02.003
- Dipak K. Dey and C. Srinivasan, Estimation of a covariance matrix under Stein’s loss, Ann. Statist. 13 (1985), no. 4, 1581–1591. MR 811511, DOI https://doi.org/10.1214/aos/1176349756
- Thomas S. Ferguson, Mathematical statistics: A decision theoretic approach, Probability and Mathematical Statistics, Vol. 1, Academic Press, New York-London, 1967. MR 0215390
- A. K. Gupta and D. K. Nagar, Matrix variate distributions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1738933
- L. R. Haff, An identity for the Wishart distribution with applications, J. Multivariate Anal. 9 (1979), no. 4, 531–544. MR 556910, DOI https://doi.org/10.1016/0047-259X%2879%2990056-3
- David A. Harville, Matrix algebra from a statistician’s perspective, Springer-Verlag, New York, 1997. MR 1467237
- W. James and Charles Stein, Estimation with quadratic loss, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I, Univ. California Press, Berkeley, Calif., 1961, pp. 361–379. MR 0133191
- Yoshihiko Konno, Improving on the sample covariance matrix for a complex elliptically contoured distribution, J. Statist. Plann. Inference 137 (2007), no. 7, 2475–2486. MR 2325450, DOI https://doi.org/10.1016/j.jspi.2006.09.022
- Yoshihiko Konno, Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss, J. Multivariate Anal. 100 (2009), no. 10, 2237–2253. MR 2560366, DOI https://doi.org/10.1016/j.jmva.2009.05.002
- Tatsuya Kubokawa, A revisit to estimation of the precision matrix of the Wishart distribution, J. Statist. Res. 39 (2005), no. 1, 97–120. MR 2195210
- T. Kubokawa and M. S. Srivastava, Robust improvement in estimation of a covariance matrix in an elliptically contoured distribution, Ann. Statist. 27 (1999), no. 2, 600–609. MR 1714715, DOI https://doi.org/10.1214/aos/1018031209
- Tatsuya Kubokawa and Muni S. Srivastava, Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data, J. Multivariate Anal. 99 (2008), no. 9, 1906–1928. MR 2466543, DOI https://doi.org/10.1016/j.jmva.2008.01.016
- Robb J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. MR 652932
- Yo Sheena, Unbiased estimator of risk for an orthogonally invariant estimator of a covariance matrix, J. Japan Statist. Soc. 25 (1995), no. 1, 35–48. MR 1347863, DOI https://doi.org/10.14490/jjss1995.25.35
- I. A. Ibragimov and M. S. Nikulin (eds.), Issledovaniya po statisticheskoĭ teorii otsenivaniya. II, “Nauka” Leningrad. Otdel., Leningrad, 1978 (Russian). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 79 (1978). MR 545117
- Akimichi Takemura, An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population, Tsukuba J. Math. 8 (1984), no. 2, 367–376. MR 767967, DOI https://doi.org/10.21099/tkbjm/1496160048
- Hisayuki Tsukuma, Improvement on the best invariant estimators of the normal covariance and precision matrices via a lower triangular subgroup, J. Japan Statist. Soc. 44 (2014), no. 2, 195–218. MR 3379071, DOI https://doi.org/10.14490/jjss.44.195
- H. R. van der Vaart, On certain characteristics of the distribution of the latent roots of asymmetric random matrix under general conditions, Ann. Math. Statist. 32 (1961), 864–873. MR 130749, DOI https://doi.org/10.1214/aoms/1177704979
References
- T. Bodnar and A. Gupta, An identity for multivariate elliptically contoured matrix distribution, Statistics & Probability Letters 79 (2009), 1327–1330. MR 2522604
- D. K. Dey and C. Srinivasan, Estimation of a covariance matrix under Stein’s loss, Ann. Statist. 13 (1985), 1581–1591. MR 811511
- T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic Press, New York, 1967. MR 0215390
- A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, CRC Press, 2000. MR 1738933
- L. R. Haff, An identity for the Wishart distribution with applications, J. Multivariate Anal. 9 (1979), 531–542. MR 556910
- D. Harville, Matrix Algebra from Statistician’s Perspective, Springer, New York, 1997. MR 1467237
- W. James and C. Stein, Estimation with quadratic loss, Proc. Fourth Berkeley Symp. Math. Statist. Prob., vol. 1, 1961, pp. 361–380. MR 0133191
- Y. Konno, Improving on the sample covariance matrix for a complex elliptically contoured distribution, Journal of Statistical Planning and Inference 7 (2007), 2475–2486. MR 2325450
- Y. Konno, Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss, J. Multivariate Anal. 100 (2009), 2237–2253. MR 2560366
- T. Kubokawa, A revisit to estimation of the precision matrix of the Wishart distribution, J. Statist. Res. 39 (2005), 91–114. MR 2195210
- T. Kubokawa and M. Srivastava, Robust improvement in estimation of a covariance matrix in an elliptically contoured distribution, Ann. Statist. 27 (1999), 600–609. MR 1714715
- T. Kubokawa and M. Srivastava, Estimation of the precision matrix of a singular Wishart distribution and its application in high dimensional data, J. Multivariate Anal. 99 (2008), 1906–1928. MR 2466543
- R. J. Muirhead, Aspects of multivariate statistical theory, Wiley, New York, 1982. MR 652932
- Y. Sheena, Unbiased estimator of risk for an orthogonal invariant estimator of a covariance matrix, J. Japan Statist. Soc. 25 (1995), no. 1, 35–48. MR 1347863
- C. Stein, Lectures on the theory of estimation of many parameters, Studies in the Statistical Theory of Estimation I (I. A. Ibraimov and M. S. Nikulin, eds.), Proceeding of Scientific Seminars of the Steklov Institute, Leningrad Division, vol. 74, 1977, pp. 4–65. MR 545117
- A. Takemura, An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population, Tsukaba J. Math. 8 (1984), 367–376. MR 767967
- H. Tsukuma, Improvement on the best invariant estimators of the normal covariance and precision matrices via a lower triangular subgroup, J. Japan Statist. Soc. 44 (2014), 195–218. MR 3379071
- H. Van der Vaart, On certain characteristics of the distribution of the latent roots of a symmetric random matrix under general conditions, Ann. Math. Statist. 32 (1961), 864–873. MR 130749
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
62H12,
62C99
Retrieve articles in all journals
with MSC (2010):
62H12,
62C99
Additional Information
G. Alfelt
Affiliation:
Department of Mathematics, Stockholm University, Roslagsvägen 101, SE-10691 Stockholm, Sweden
Email:
gustava@math.su.se
Keywords:
Random matrices,
matrix-variate gamma distribution,
decision theory
Received by editor(s):
June 21, 2018
Published electronically:
February 27, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
