Abstract
The problems of estimating ratio of scale parameters of two distributions with unknown location parameters are treated from a decision-theoretic point of view. The paper provides the procedures improving on the usual ratio estimator under strictly convex loss functions and the general distributions having monotone likelihood ratio properties. In particular,double shrinkage improved estimators which utilize both of estimators of two location parameters are presented. Under order restrictions on the scale parameters, various improvements for estimation of the ratio and the scale parameters are also considered. These results are applied to normal, lognormal, exponential and pareto distributions. Finally, a multivariate extension is given for ratio of covariance matrices.
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References
Arnold, B. C. (1970). Inadmissibility of the usual scale estimate for a shifted exponential distribution,J. Amer. Statist. Assoc.,65, 1260–1264.
Brewster, J. F. (1974). Alternative estimators for the scale parameter of the exponential distribution with unknown location,Ann. Statist.,2, 553–557.
Brewster, J. F. and Zidek, J. V. (1974). Improving on equivariant estimators,Ann. Statist.,2, 21–38.
Brown, L. D. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters,Ann. Math. Statist.,39, 29–48.
Gelfand, A. E. and Dey, D. K. (1988). On the estimation of a variance ratio,J. Statist. Plann. Inference,19, 121–131.
Goutis, C. and Casella, G. (1991). Improved invariant confidence intervals for a normal variance,Ann. Statist.,19, 2015–2031.
Kubokawa, T. (1994). A unified approach to improving equivariant estimators,Ann. Statist. (to appear).
Kubokawa, T. and Saleh, A. K. Md. E. (1993). On improved positive estimators of variance components,Statist. Decisions (to appear).
Kubokawa, T., Robert, C. and Saleh, A. K. Md. E. (1992). Empirical Bayes estimation of the covariance matrix of a normal distribution with unknown mean under an entropy loss,Sankhyā Ser. A,54, 402–410.
Kubokawa, T., Robert, C. P. and Saleh, A. K. Md. E. (1993a). Estimation of noncentrality parameters,Canad. J. Statist.,21, 45–57.
Kubokawa, T., Honda, T., Morita, K. and Saleh, A. K. Md. E. (1993b). Estimating a covariance matrix of a normal distribution with unknown mean,J. Japan Statist. Soc.,23, 131–144.
Maatta, J. M. and Casella, G. (1990). Developments in decision-theoretic variance estimation,Statist. Sci.,5, 90–120.
Madi, M. and Tsui, K.-W. (1990). Estimation of the ratio of the scale parameters of two exponential distributions with unknown location parameters,Ann. Inst. Statist. Math.,42, 77–87.
Nagata, Y. (1989). Improvements of interval estimations for the variance and the ratio of two variance,J. Japan Statist. Soc.,19, 151–161.
Nagata, Y. (1991). Improved interval estimation for the scale parameter of exponential distribution with an unknown location-scale parameter,Quality, Journal of the Japanese Society of Quality Control,21, 5–10 (in Japanese).
Perron, F. (1990). Equivariant estimators of the covariance matrix,Canad. J. Statist.,18, 179–182.
Sinha, B. K. and Ghosh, M. (1987). Inadmissibility of the best equivariant estimators of the variance-covariance matrix, the precision matrix, and the generalized variance under entropy loss,Statist. Decisions,5, 201–227.
Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean,Ann. Inst. Statist. Math.,16, 155–160.
Takeuchi, K. (1991). Personal communications.
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Kubokawa, T. Double shrinkage estimation of ratio of scale parameters. Ann Inst Stat Math 46, 95–116 (1994). https://doi.org/10.1007/BF00773596
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DOI: https://doi.org/10.1007/BF00773596