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ON CONSTRUCTION OF IMPROVED ESTIMATORS IN

MULTIPLE-DESIGN MULTIVARIATE LINEAR MODELS

UNDER GENERAL RESTRICTION

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T. SHIRAISHI^{1} AND Y. KONNO^{2}

^{1} *Department of Mathematical Sciences, Yokohama City
University,*

22-2 Seto, Kanazawa-ku, Yokohama 236, Japan

^{2} *Department of Mathematics and Informatics, Chiba
University,*

1-33 Yayoi-cho, Chiba 263, Japan

(Received December 20, 1993; revised January 4, 1995)

**Abstract.**
Consider a set of *p* equations **Y**_{i} =
**X**_{i} *xi*_{i}+*epsilon*_{i}, *i* = 1, ···, *p*, where
the rows of the random error matrix *epsilon*_{1}, ···,
*epsilon*_{p} : *n* × *p* are mutually independent and
identically distributed with a *p*-variate distribution function
*F*(**x**) having null mean and finite positive definite
variance-covariance matrix **Sigma**. We are mainly interested
in an improvement upon a feasible generalized least squares
estimator (FGLSE) for *xi* = (*xi*'_{1}, ···,
*xi*'_{p})' when it is a priori suspected that
**C***xi* = **c**_{0} may hold. For this problem, Saleh and
Shiraishi (1992, *Nonparametric Statistics and Related
Topics* (ed. A. K. Md. E. Saleh), 269-279, North-Holland,
Amsterdam) investigated the property of estimators such as the
shrinkage estimator (SE), the positive-rule shrinkage estimator
(PSE) in the light of their asymptotic distributional risks
associated with the Mahalanobis loss function. We consider a general
form of estimators and give a sufficient condition for proposed
estimators to improve on FGLSE with respect to their asymptotic
distributional quadratic risks (ADQR). The relative merits of
these estimators are studied in the light of the ADQR under local
alternatives. It is shown that the SE, the PSE and the Kubokawa-type
shrinkage estimator (KSE) outperform the FGLSE and that the PSE is
the most effective among the four estimators considered under
**C***xi* = **c**_{0}. It is also observed that the PSE and the
KSE fairly improve over the FGLSE. Lastly, the construction of
estimators improved on a generalized least squares estimator is
studied, assuming normality when **Sigma** is known.

*Key words and phrases*:
Shrinkage estimators,
generalized least squares estimators, asymptotic distribution,
seemingly unrelated regression model.

**Source**
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