Abstract
In this paper, we give an ever wider and new class of minimax estimators for the location vector of an elliptical distribution (a scale mixture of normal densities) with an unknown scale parameter. The its application to variance reduction for Monte Carlo simulation when control variates are used is considered. The results obtained thus extend (i) Berger's result concerning minimax estimation of location vectors for scale mixtures of normal densities with known scale parameter and (ii) Strawderman's result on the estimation of the normal mean with common unknown variance.
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References
Baranchik, A. J. (1973). A family of minimax estimators of the mean of a multivariate normal distribution. Ann. Math. Statist., 41, 642–645.
Bauer, K. W. and Wilson, J. (1989). Control-variate selection criteria, SMS 89–29, Department of Statistics and School of Industrial Engineering, Purdue University, Indiana.
Berger, J. (1975). Minimax estimation of location vectors for wide class of densities, Ann. Statist., 10, 81–92.
Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis, Springer, New York.
Brandwein, A. C. and Strawderman, W. E. (1990). Stein estimation: the spherically symmetric case, Statist. Sci., 5, 358–368.
Bravo, G. and MacGibbon, B. (1988). Improved shrinkage estimators for the mean vector of a scale mixture of normals with unknown variance, Canad. J. Statist., 16, 237–245.
Bravo, G. and MacGibbon, B. (1990). Improved estimators of a location vector with unknown scale parameter. Comm. Statist. Theory Methods, 19, 3657–3670.
Brown, L. D. (1990). An ancillary paradox which appears in multiple regression. Ann. Statist., 18, 471–493.
Cellier, D., Fourdrinier, D. and Robert, C. (1989). Robust shrinkage estimators of the location parameter for elliptically symmetric distributions, J. Multivariate Anal., 29, 39–52.
DasGupta, A. and Rubin, H. (1988). Bayesian estimation subject to minimaxity of the mean of a multivariate normal distribution in the case of a common unknown variance: a case for Bayesian robustness, Statistical Decisions & Related Topics (eds. J.Berger and S.Gupta), Springer, New York.
Fishman, G. S. (1989). Monte Carlo, control variates, and stochastic ordering, SIAM J. Sci. Statist. Comput., 10, 187–204.
Gleser, L. J. and Tan, M. (1989). Minimax estimation of location vectors in elliptical distributions with unknown scale parameter, Mimeo. Series, # 89–37, Department of Statistics, Purdue University, Indiana.
Hwang, J. T. and Casella, G. (1982). Minimax confidence sets for the mean vector of a multivariate normal distribution, Ann. Statist., 10, 868–881.
Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter generalization, Sankhyā Ser. A, 32, 419–430.
Lavenberg, S. S. and Welch, P. D. (1981). A perspective on the use of control variates to increase the efficiency of Monte Carlo simulation, Management Sci., 27, 332–335.
Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, Wiley, New York.
Nelson, B. L. (1987). A perspective on variance reduction in dynamic simulation experiments, Comm, Statist. Simulation Comput., 16, 385–426.
Stein, C. (1960). Multiple regression, Contributions to Probability and Statistics, Essays in Honor of Harold Hotelling (eds. I.Olkin, S. G.Ghurye, W.Hoeffding, W. G.Madow and H. B.Mann), Stanford University Press, California.
Strawderman, W. E. (1973). Proper Bayes minimax estimators of the multivariate normal mean vector for the case of common unknown variances, Ann. Statist., 1, 1189–1194.
Takada, Y. (1979). A family of minimax estimators in some multiple regression problems, Ann. Statist., 7, 1144–1147.
Tan, M. (1990). Shrinkage, GMANOVA, control variates and their applications, Ph.D. Thesis, Department of Statistics, Purdue University, Indiana.
Wilson, J. R. (1984). Variance reduction techniques for digital simulation, Mathematics and Management Science, 1, 227–312.
Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate student t-error terms, J. Amer. Statist. Assoc., 71, 400–405.
Zidek, J. (1978). Deriving unbiased risk estimators of multinormal mean and regression coefficient estimators using zonal polynomials, Ann. Statist., 6, 769–782.
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Research partially supported by National Science Foundation, Grant #DMS 8901922.
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Tan, M., Gleser, L.J. Minimax estimators for location vectors in elliptical distributions with unknown scale parameter and its application to variance reduction in simulation. Ann Inst Stat Math 44, 537–550 (1992). https://doi.org/10.1007/BF00050704
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DOI: https://doi.org/10.1007/BF00050704