×

The bias and risk functions of some Stein-rules in elliptically contoured distributions. (English) Zbl 1284.62191

Summary: In this paper, we derive the bias and risk functions of a class of shrinkage estimators of several mean parameter matrices of matrix-variate elliptically contoured distributions. More specifically, we generalize some recent findings in three ways. First, the class of distributions under consideration is more general than the Gaussian distribution case, which is often studied in literature. Second, the uncertain subspace candidate is more general than that considered in literature. Finally, we generalize some recent identities, which are useful in establishing the risk and the bias of matrix shrinkage estimators.

MSC:

62F30 Parametric inference under constraints
62J07 Ridge regression; shrinkage estimators (Lasso)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Batsidis, ”Robustness of the Likelihood Ratio Test for Detection and Estimation of aMean Change Point in a Sequence of Elliptically Contoured Observations”, Statistics: A J. Theoret. and Appl. Statist. 44(1), 17–24 (2010). · Zbl 1282.62143
[2] A. Batsidis and K. Zografos, ”Statistical Inference for Location and Scale of Elliptically Contoured Models with Monotone Missing Data”, J. Statist. Plann. and Inference 136(8), 2606–2629 (2006). · Zbl 1090.62054 · doi:10.1016/j.jspi.2004.10.021
[3] A. Batsidis, N. Martin, L. Pardo, and K. Zografos (2012). ”A Necessary Power Divergence-Type Family of Tests for Testing Elliptical Symmetry”, J. Statist. Comput. and Simulation,” DOI:10.1080/00949655.2012.694437 · Zbl 1302.62131
[4] K. C. Chu, ”Estimation and Decision for Linear Systems with Elliptical Random Processes”, IEEE Trans. Auto. Cont. 18, 499–505 (1973). · Zbl 0263.93049 · doi:10.1109/TAC.1973.1100374
[5] A. J. Díaz-García and G. Gonzclez-Farías, Singular Matrix Variate Skew-Elliptical Distribution and the Distribution of General Linear Transformation, Comunicación Técnica No I-05-03 (PE/CIMAT) (2005).
[6] A. K. Gupta and T. Varga, ”Normal Mixture Representations of Matrix Variate Elliptically Contoured Distributions”, Sankhyā, 57, 68–78 (1995). · Zbl 0857.62050
[7] A. K. Gupta and T. Varga, ”A New Class of Matrix Variate Elliptically Contoured Distributions”, J. ltal. Statist. Soc., No. 2, 255–270 (1994). · Zbl 1446.62138
[8] F. Huffer and C. Park, ”A Test for Elliptical Symmetry”, J. Multivar. Anal. 98, 256–281 (2007). · Zbl 1105.62063 · doi:10.1016/j.jmva.2005.09.011
[9] A. J. Izenman, Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning, in Springer Texts in Statistics (New York, Springer, 2008). · Zbl 1155.62040
[10] G. G. Judge and M. E. Bock, The Statistical Implication of Pre-Test and Stein-Rule Estimators in Econometrics (Amsterdam, North Holland, 1978). · Zbl 0395.62078
[11] K. Jain, S. Singh, and S. Sharma, ”Restricted Estimation in Multivariate Measurement Error Regression Model”, J. Multivar. Anal. 102, 264–280 (2011). · Zbl 1327.62403 · doi:10.1016/j.jmva.2010.09.004
[12] W. James and C. Stein, ”Estimation with Quadratic Loss”, in Proc. Fourth Berkeley Symp. Math. Statist. Probab., Berkeley (Univ. of California Press, 1961), pp. 361–379. · Zbl 1281.62026
[13] T. Kollo and D. V. Rosen, Advanced Multivariate Statistics with Matrices (Mathematics and Its Applications) (Springer, Dordrecht, 2010).
[14] J. Liang, K.-T. Fang, and F. J. Hickernell, ”Some Necessary Uniform Tests for Spherical Symmetry”, Ann. Inst. Statist. Math. 60, 679–696 (2008). · Zbl 1169.62052 · doi:10.1007/s10463-007-0121-9
[15] S. Nkurunziza, ”The Risk of Pretest and Shrinkage Estimators”, Statistics: A J. Theoret. and Appl. Statist. 46(3), 305–312 (2012a). · Zbl 1241.62095
[16] S. Nkurunziza, ”Shrinkage Strategies in Some Multiple Multi-Factor Dynamical Systems”, ESAIM: Probab. and Statist. 16, 139–150 (2010b). · Zbl 1302.62184 · doi:10.1051/ps/2010015
[17] C. R. Rao, H. Toutenburg, S. Shalabh, and C. Heumann, Linear Models and Generalizations: Least Squares and Alternatives, 3d ed. (Springer, Berlin-Heidelberg, 2008). · Zbl 1151.62063
[18] L. Sakhanenko, ”Testing for Ellipsoidal Symmetry: A Comparison Study”, Comput. Statist. and Data Analysis 53, 565–581 (2008). · Zbl 1301.62056 · doi:10.1016/j.csda.2008.08.029
[19] R. S. Singh, ”Estimation of Error Variance in Linear Regression Models with Errors Having Multivariate Student-t Distribution with Unknown Degrees of Freedom”, Economic Letters 27, 47–53 (1988). · Zbl 1328.62439 · doi:10.1016/0165-1765(88)90218-2
[20] R. S. Singh, ”James-Stein Rule Estimators in Linear Regression Models with Multivariate t Distributed Error”, Australian J. Statist. 33, 145–158 (1991). · Zbl 0781.62107 · doi:10.1111/j.1467-842X.1991.tb00422.x
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.