Admissible Estimators in the General Multivariate Linear Model with Respect to Inequality Restricted Parameter Set
© Shangli Zhang et al. 2009
Received: 28 May 2009
Accepted: 11 August 2009
Published: 27 August 2009
By using the methods of linear algebra and matrix inequality theory, we obtain the characterization of admissible estimators in the general multivariate linear model with respect to inequality restricted parameter set. In the classes of homogeneous and general linear estimators, the necessary and suffcient conditions that the estimators of regression coeffcient function are admissible are established.
Throughout this paper, , and denote the set of real matrices, the subset of consisting of symmetric matrices, and the subset of consisting of nonnegative definite matrices, respectively. The symbols , and stand for the transpose, the range, Moore-Penrose inverse, generalized inverse, and trace of , respectively. For any , means .
Consider the general multivariate linear model with respect to inequality restricted parameter set:
where is an observable random matrix, , , and are known matrices, respectively. , ( ) are unknown matrices. is the error matrix. denotes the vector made of the columns of and denotes the Kronecker product.
and there exists , such that , then is said to be better than . If there does not exist any estimator in set that is better than , where parameters , then is called the admissible estimator of in the set . We denote it by .
In the case of , model (1.1) degenerates to the general multivariate linear model without restrictions. Under the quadratic loss function, many articles discussed the admissibility of linear estimators, such as Cohen , Rao , LaMotte , etc. Under the matrix loss function, Zhu and Lu  and Baksalary and Markiewicz  studied the admissibility of linear estimators when respectively. Deng et al.  discussed the admissibility under the matrix loss in multivariate model. Markiewicz  discussed the admissibility in the general multivariate linear model. Marquardt  and Perlman  pointed out that the least square estimator is not still the admissible estimator if the parameters are restricted. Further, Groß and Markiewicz  pointed out that the admissible linear estimator has the form of ridge estimator if the parameters have no restrictions. Therefore, it is useful and important to discuss the admissibility of linear estimators when the parameters have some restrictions.
Zhu and Zhang , Lu , Deng and Chen  studied the admissibility of linear estimators under the quadratic loss and matrix loss when . Qin et al.  studied the admissibility of the estimators of estimable function under the loss function in multivariate linear model with respect to restricted parameter set when . In their case, whether an estimator is better than another or not does not depend on the regression parameters. It is easy to generalize the conclusions from univariate linear model to multivariate linear model. However under the matrix loss (1.2), it is more complicated. In this case, whether an estimator is better than another depends on the regression parameters.
In this paper, using the methods of linear algebra and matrix theory, we discuss the admissibility of linear estimators in model (1.1) under the matrix loss (1.2). We prove that the admissibility of the estimators of estimable function under univariate linear model and multivariate linear model are equivalent in the class of homogeneous linear estimators, and some sufficient and necessary conditions that the estimators in the general multivariate linear model with respect to restricted parameter set are admissible are obtained whether the function of parameter is estimable or not, which enriches the theory of admissibility in multivariate linear model.
2. Main Results
and the two equalities above cannot hold simultaneously.
Therefore, (2.7) holds. It is obvious that the two equalities in (2.6) and (2.7) cannot hold simultaneously.
Consider univariate linear model with respect to restricted parameter set:
Similarly to Lemma 2.2, we have the following lemma.
and the two equalities above cannot hold simultaneously.
From Lemmas 2.2 and 2.3, we need only to prove the equivalence of (2.7) and (2.13).
The claim follows.
From this Theorem, we can easily generalize the result under univariate linear model to the case under multivariate linear model in the class of homogeneous linear estimators.
From the corresponding theorem in article Deng and Chen , under the model (2.10) with the loss function (2.11), if is estimable, then if and only if (1) and (2) in Theorem 2.6 are satisfied. Now Theorem 2.6 follows from Theorem 2.4.
Therefore, there exists , for , the right side of (2.25) is nonnegative definite and its rank is . If is small enough, for every , we have , and the equality cannot always hold if (2) does not hold. It contradicts .
The proof is complete.
Similarly to Theorem 2.14, we have the following theorem.
The authors would like to thank the Editor Dr. Kunquan Lan and the anonymous referees whose work and comments made the paper more readable. The research was supported by National Science Foundation (60736047, 60772036, 10671007) and Foundation of BJTU (2006XM037), China.
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