×
Arashi, M.; Tabatabaey, S. M. M.

Improved variance estimation under sub-space restriction. (English) Zbl 1163.62041

J. Multivariate Anal. 100, No. 8, 1752-1760 (2009).
Summary: For the linear regression model \(y = X \beta +\varepsilon\), we assume that for a given positive definite scale matrix \(\varSigma\), the error vector has a multivariate normal distribution and \(\varSigma\) has the inverted Wishart distribution. Under an orthogonal sub-space restriction \(H\beta = h\), we propose restricted unbiased, preliminary test and C. Stein-type [Ann. Inst. Stat. Math. 16, 155–160 (1964; Zbl 0144.41405)] estimators of the variance of the error term, when the scale of the inverse Wishart distribution is assumed to be unknown. We compare the weighted quadratic risks of the underlying estimators and propose dominance pictures for them.

Cited in 12 Documents

MSC:

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
62F30 Parametric inference under constraints

Keywords:

multivariate Student-\(t\) distribution; inverse Wishart distribution; restricted estimator; preliminary test estimator; Stein-type estimator

Citations:

Zbl 0144.41405
Cite Review PDF
Full Text: DOI

References:

[1] Arashi, M.; Tabatabaey, S. M.M., Comparing risks of estimators in multiple regression model under multivariate-Student t errors, J. Iran. Stat. Sci., 1, 95-108 (2007), (in Persian)
[2] Nadarajah, S.; Kotz, S., Mathematical properties of the multivariate t distribution, Acta Appl. Math., 89, 53-84 (2005) · Zbl 1092.62060
[3] Ullah, A.; Walsh, V. Z., On the robustness of LM, LR and W tests in regression models, Econometrica, 52, 1055-1066 (1984) · Zbl 0564.62025
[4] Lange, K. L.; Little, R. J.A.; Taylor, J. M.G., Robust statistical modeling using the t-distribution, J. Amer. Statist. Assoc., 84, 881-896 (1989)
[5] S.M.M. Tabatabaey, Preliminary test approach estimation: Regression model with spherically symmetric errors, Ph.D. Thesis, Carleton University, Canada, 1995; S.M.M. Tabatabaey, Preliminary test approach estimation: Regression model with spherically symmetric errors, Ph.D. Thesis, Carleton University, Canada, 1995
[6] Arashi, M.; Tabatabaey, S. M.M., Stein-type improvement under stochastic constraints: Use of multivariate Student-t model in regression, Statist. Probab. Lett., 78, 2142-2153 (2008) · Zbl 1283.62140
[7] Loschi, R. H.; Iglesian, P. L.; Arellano-Valle, R. B., Predictivistic characterization of multivariate Student-t model, J. Multivariate Anal., 85, 10-23 (2003) · Zbl 1013.62061
[8] Saleh, A. K.Md. E., Theory of Preliminary Test and Stein-Type Estimation with Applications (2006), John Wiley: John Wiley New York · Zbl 1094.62024
[9] Arellano-Valle, R. B.; Bolfarine, H., On some characterization of the t-distribution, Statist. Probab. Lett., 25, 79-85 (1995) · Zbl 0838.62040
[10] Zellner, A., Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms, J. Amer. Statist. Assoc., 71, 400-405 (1976) · Zbl 0348.62026
[11] Clarke, J. A.; Giles, D. E.A.; Wallace, D., Preliminary-test estimation of the error variance in linear regression, Econometric Theory, 3, 299-304 (1987) · Zbl 0611.62076
[12] Bancroft, T. A., On biases in estimation due to the use of preliminary test of significance, Annal. Math. Stat., 15, 195-204 (1944) · Zbl 0063.00180
[13] Giles, J. A., Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances, J. Econometrics, 50, 377-398 (1991) · Zbl 0745.62067
[14] Giles, J. A., Estimation of the error variance after a preliminary-test of homogeneity in a regression model with spherically symmetric disturbances, J. Econometrics, 53, 345-361 (1992) · Zbl 0746.62063
[15] Judge, W.; Bock, M. E., The Statistical Implication of Pre-Test and Stein-Rule Estimators in Econometrics (1978), North-Holland: North-Holland New York · Zbl 0395.62078
[16] Chu, K. C., Estimation and decision for linear systems with elliptically random process, IEEE Trans. Automat. Control, 18, 499-505 (1973) · Zbl 0263.93049
[17] Maatta, J. M.; Casella, G., Developments in decision-theoretic variance estimation, Statist. Sci., 5, 90-120 (1990) · Zbl 0955.62529
[18] Arashi, M.; Tabatabaey, S. M.M.; Khan, S., Estimation in multiple regression model with elliptically contoured errors under MLINEX loss, J. Appl. Probab. Statist., 3, 23-35 (2008) · Zbl 1302.62163
[19] Stein, C., Inadmissibility of the usual estimate of the variance of a normal distribution with unknown mean, Ann. Inst. Statist. Math., 16, 155-160 (1964) · Zbl 0144.41405
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.