Admissibility in general linear model with respect to an inequality constraint under balanced loss
© Zhang and Gui; licensee Springer. 2014
Received: 3 September 2013
Accepted: 30 January 2014
Published: 13 February 2014
Since Zellner (Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions, pp. 377-390, 1994) proposed the balanced loss function, many researchers have been attracted to the field concerned. In this paper, under a generalized balanced loss function, we investigate the admissibility of linear estimators of the regression coefficient in general Gauss-Markov model (GGM) with respect to an inequality constraint. The necessary and sufficient conditions that the linear estimators of regression coefficient function are admissible are established, in the class of homogeneous/inhomogeneous linear estimation, respectively.
Throughout this paper, the symbols , , , , and stand for the transpose, the range, Moore-Penrose inverse, generalized inverse, rank, and trace of matrix A, respectively.
where y is a observable random vector. X is an known design matrix and , ε is a random error vector. β and are unknown parameters.
where , S is a known positive definite matrix. The balanced loss function takes both the precision of the estimator and the goodness-of-fit of the model into account. Compared to the standards in (1.2) and (1.3), it is a more comprehensive one that measures the estimate.
Much work has been done on the parameter estimation under the balanced loss function. [2–5] studied the risk function of some specific estimators. [6–8] did some work on the application of the balanced loss function. [9–13] investigated the goodness of the estimators under the balanced loss function.
where , S is a known matrix.
where r is a known vector. If , then the constraint condition always holds. This model embraces the unconstraint case.
and there exists , such that , where the risk function , then is said to be better than . If there does not exist any estimator in set Ξ that is better than , where parameters β and σ take values in T, then is called the admissible estimator of Kβ in the set Ξ. We denote it by .
where HL is the class of homogeneous linear estimators and L is the class of inhomogeneous linear estimators.
The admissibility is the most basic and influential rationality requirement of classical statistical decision theory. When the parameters are unconstrained, comprehensive results have been obtained. For instance, [14–17]etc. studied the admissibility in univariate linear model. As [18, 19] pointed out, when the parameters are constrained, the least square estimator may not be admissible. So it is significant to discuss the admissibility of linear estimator in linear model with some constraints. For the Gauss-Markov model with constraints,  developed the admissible estimator. Some other researchers dedicated to this study. [21–24] studied the admissibility in the linear model with an ellipsoidal constraint. For the linear model with an inequality constraint, [25–28] studied the admissibility of linear estimator of parameters in the univariate and multivariate linear models under the quadratic and matrix loss, respectively. However, under the balanced loss, the model with an inequality constraint has not been considered.
2 Admissibility in the class of homogeneous linear estimators
where is an estimator of .
where , .
where . Notice that , we have .
and the equality holds if and only if . □
Remark 2.1 This lemma indicates the class of estimators is a complete class of HL. That is, for any estimator δ not in , there exists an estimator in such that is better than δ.
Proof The lemma can easily be verified from (2.4). □
Lemma 2.4 Consider the model (1.5) with the loss function (1.6), suppose , then if and only if in model (2.6) with the loss function (2.7), where .
Equations (2.10), (2.11) and Lemma 2.3 indicate that if there exists an estimator of β, is better than , then , the estimator of Cβ, is better than . □
Lemma 2.5 Consider the model (2.6) with the loss function (2.7), holds if and only if under the quadratic loss, holds.
Proof The proof is straightforward. We omit the details. □
Theorem 2.1 Consider the model (2.6) with the loss function (2.7), if and only if .
Proof The necessity is trivial. We only need to prove the sufficiency. For any , if there exists that is better than AY, by Lemma 2.3, for any , (2.8) and (2.9) hold. Notice that (2.9) still holds if replacing β by −β. In other words, for any , (2.9) holds. Since , thus, for any , (2.8) and (2.9) hold. It contradicts with . □
Proof By Lemma 2.2, (1) holds. Further, is equivalent to . By Lemma 2.4, in the model (1.5) with the loss (1.6), holds if and only if in model (2.6) with the loss (2.7), where . It is also equivalent to in model (2.6) with the loss (2.1) by Lemma 2.5. Therefore, when the condition (1) is satisfied, according to Lemma 2.1 and simple computations, we have holds if and only if (2) and (3) are satisfied. □
Remark 2.2 The following example indicates that the conditions in the above theorem can be satisfied.
For the diagonal matrix , we consider the admissibility of Ay. The condition (1) in Theorem 2.2 is satisfied. Theorem 2.2(3) implies that . Theorem 2.2(2) implies that . Thus, only if and , Ay is an admissible estimate of β.
3 Admissibility in the class of inhomogeneous linear estimators
In this section, we study the admissibility in the class of inhomogeneous linear estimators.
if and only if and , where is the dual cone of C.
Proof This lemma can be found in . □
and the equality holds if and only if , . This means if , then is better than . It is a contradiction.
is better than , which contradicts .
It implies that no estimator is better than AY. Thus, . □
In fact, the converse part of Theorem 3.1 is also true. We present this in the following theorem.
Proof By the proof of (1) in Theorem 3.1, we need to prove that there does not exist matrix and such that is better than , where .
Plug (3.9), (3.10), and (3.14) into (3.7) and we find that the equality in (3.7) holds. It means there does not exist an estimator that is better than . Therefore, holds. □
We summarize Theorem 2.2 and Theorem 3.2 in the following theorem.
In this paper, under a generalized balanced loss function, we study the admissibility of linear estimators of the regression coefficient in general Gauss-Markov model with respect to an inequality constraint. The necessary and sufficient conditions that the linear estimators of regression coefficient function are admissible are obtained, in the class of homogeneous and inhomogeneous linear estimation, respectively.
This work was partially supported by National Natural Science Foundation of China (61070236, U1334211, 11371051) and the Project of State Key Laboratory of Rail Traffic Control and Safety (RCS2012ZT004), Beijing Jiaotong University.
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