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DOUBLE SHRINKAGE ESTIMATION OF RATIO

OF SCALE PARAMETERS

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TATSUYA KUBOKAWA

*Department of Economics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan*
(Received November 9, 1992; revised May 26, 1993)

**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.

*Key words and phrases*:
Point estimation, ratio of variances,
shrinkage estimation, inadmissibility, Stein's truncated rule, monotone
likelihood ratio property, normal, exponential, noncentral chi-square
distributions, ratio of covariance matrices.

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