Abstract
This paper discusses admissibilities of estimators in a class of linear models, which include the following common models: the univariate and multivariate linear models, the growth curve model, the extended growth curve model, the seemingly unrelated regression equations, the variance components model, and so on. It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms, called as Model II, are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function. It is also proved under Model II that the usual estimators of β are admissible for p ⩽ 2 with a quadratic loss function, and are admissible for any p with a matrix loss function, where p is the dimension of β.
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References
Brandwein A R C, Strawderman W E. Minimax estimation of location parameter for spherically symmetric unimodal distributions under quadratic loss. Ann Statist, 1978, 6: 377–416
Brown L D. On the admissibility of invariant estimators of one or more location parameters. Ann Math Statist, 1966, 37: 1087–1136
Chen X R, Chen G J, Wu Q G, et al. Theory of Estimation of Parameters in Linear Models (in Chinese). Beijing: Science Press, 1985
Cheng P. Admissibility of simultaneous estimation of several parameters. J Systems Sci Math Sci, 1982, 2: 176–195
Cohen A. All admissible linear estimates of the mean vector. Ann Math Statist, 1966, 37: 458–463
Fama E F. The behaviour of stock market prices. J Business, 1965, 38: 34–105
James W, Stein C. Estimation with quadratic loss. In: Proceedings of Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1. Berkeley: University of California Press, 1961, 361–379
Lange K L, Little R J A, Taylor J M G. Robust statistical modeling using the t distribution. J Amer Statist Assoc, 1989, 84: 881–896
Richardson M, Smith T. A test for multivariate normality in stock returns. J Business, 1993, 66: 295–321
Kotz S, Nadarajah S. Multivariate t Distributions and Their Applications. Cambridge: Cambridge University Press, 2004
Stein C. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In: Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1. Berkeley: University of California Press, 1956, 197–206
Wu Q G. Several results on admissibility of linear estimator of stochastic regression coefficients and parameters (in Chinese). Acta Math Appl Sinica, 1988, 11: 95–106
Wu Q G, Chen J B. All admissible linear estimates of regression coefficients under matrix loss. J Systems Sci Math Sci, 1989, 21: 80–91
Wu Q G, Noda K. Admissibilities of matrix linear estimators in multivariate linear models. J Statist Plann Inference, 2006, 136: 3852–3870
Xu X Z, Wu Q G. NS condition of admissibility for the linear estimator of normal mean with unknown variance. Acta Math Sin (Engl Ser), 2005, 21: 1083–1086
Yang G Q, Wu Q G. Existence conditions for the uniformly minimum risk unbiased estimators in a class of linear models. J Multivariate Anal, 2004, 88: 76–88
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Yang, G., Wu, Q. Admissibilities of linear estimator in a class of linear models with a multivariate t error variable. Sci. China Math. 53, 2011–2019 (2010). https://doi.org/10.1007/s11425-010-4050-3
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DOI: https://doi.org/10.1007/s11425-010-4050-3