On the Weighted Mixed Almost Unbiased Ridge Estimator in Stochastic Restricted Linear Regression

Abstract

We introduce the weighted mixed almost unbiased ridge estimator (WMAURE) based on the weighted mixed estimator (WME) (Trenkler and Toutenburg 1990) and the almost unbiased ridge estimator (AURE) (Akdeniz and Erol 2003) in linear regression model. We discuss superiorities of the new estimator under the quadratic bias (QB) and the mean square error matrix (MSEM) criteria. Additionally, we give a method about how to obtain the optimal values of parameters 𝑘 and 𝑤. Finally, theoretical results are illustrated by a real data example and a Monte Carlo study.

1. Introduction

Consider the linear regression model 𝑦=𝑋𝛽+𝜀, (1) where 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇 is an 𝑛-dimensional response vector, 𝑋=(𝑋₁,𝑋₂,…,𝑋_𝑛)^𝑇 with 𝑋_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑝)^𝑇 is a known 𝑛×𝑝 matrix of full column rank, 𝛽 is a 𝑝×1 vector of unknown parameters, and 𝜀 is an 𝑛×1 vector of errors with expectation 𝐸(𝜀)=0 and covariance matrix Cov(𝜀)=𝜎²𝐼_𝑛,𝐼_𝑛 is an identity matrix of order 𝑛×𝑛.

It is well known that the ordinary least squares estimator (LS) for 𝛽 is given by 𝛽_LS=𝑋^𝑇𝑋⁻¹𝑋^𝑇𝑦, (2) which has been treated as the best estimator for a long time. However, many results have proved that LS is no longer a good estimator when the multicollinearity is present in model (1). To tackle this problem, some suitable biased estimators have been developed, such as principal component regression estimator (PCR) [1], ordinary ridge estimator (RE) [2], 𝑟−𝑘 class estimator [3], Liu estimator (LE) [4], and 𝑟−𝑑 class estimator [5]. Kadiyala [6] introduced a class of almost unbiased shrinkage estimator which can be not only almost unbiased but also more efficient than the LS. Singh et al. [7] introduced the almost unbiased generalized ridge estimator by the jackknife procedure, and Akdeniz and Kaçiranlar [8] studied the almost unbiased generalized Liu estimator. By studying bias corrected estimators of the RE and the LE, Akdeniz and Erol [9] discussed the almost unbiased ridge estimator (AURE) and the almost unbiased Liu estimator (AULE).

An alternative technique to tackle the multicollinearity is to consider the parameter estimator in addition to the sample information, such as some exact or stochastic restrictions on unknown parameters. When additional stochastic linear restrictions on unknown parameters are assumed to be held, Durbin [10], Theil and Goldberger [11], and Theil [12] proposed the ordinary mixed estimator (OME). Hubert and Wijekoon [13] proposed the stochastic restricted Liu estimator, and Yang and Xu [14] obtained a new stochastic restricted Liu estimator. By grafting the RE into the mixed estimation procedure, Li and Yang [15] introduced the stochastic restricted ridge estimator. When the prior information and the sample information are not equally important, Schaffrin and Toutenburg [16] studied the weighted mixed regression and developed the weighted mixed estimator (WME). Li and Yang [17] grafted the RE into the weighted mixed estimation procedure and proposed the weighted mixed ridge estimator (WMRE).

In this paper, by combining the WME and the AURE, we propose a weighted mixed almost unbiased ridge estimator (WMAURE) for unknown parameters in a linear regression model when additional stochastic linear restriction is supposed to be held. Furthermore, we discuss the performance of the new estimator over the LS, WME, AURE, and WMRE with respect to the quadratic bias (QB) and the mean square error matrix (MSEM) criteria.

The rest of the paper is organized as follows. In Section 2, we describe the statistical model and propose the weighted mixed almost unbiased ridge estimator. We compare the new estimator with the weighted mixed ridge estimator and the almost unbiased ridge estimator under the quadratic bias criterion in Section 3. In Section 4, superiorities of the proposed estimator over relative estimators are considered under the mean square error matrix criterion. In Section 5, the selection of parameters 𝑘 and 𝑤 is discussed. Finally, to justify the superiority of the new estimator, we perform a real data example and a Monte Carlo simulation study in Section 6. We give some conclusions in Section 7.

2. The Proposed Estimator

The ordinary ridge estimator proposed by Hoerl and Kennard [2] is defined as 𝛽_RE(𝑘)=(𝑆+𝑘𝐼)⁻¹𝑋^𝑇𝑦=𝑆_𝑘⁻¹𝑋^𝑇𝑦, (3) where 𝑆_𝑘=𝑆+𝑘𝐼, 𝑘>0. Let 𝑇_𝑘=(𝐼+𝑘𝑆⁻¹)⁻¹=𝑆_𝑘⁻¹𝑆=𝑆𝑆_𝑘⁻¹; we may rewrite 𝛽_RE(𝑘) as 𝛽_RE(𝑘)=𝑇_𝑘𝛽_LS=𝑇_𝑘𝑆⁻¹𝑋^𝑇𝑦=𝑆⁻¹𝑇_𝑘𝑋^𝑇𝑦. (4)

The almost unbiased ridge estimator obtained by Akdeniz and Erol [9] is denoted as 𝛽_AURE(𝑘)=𝐼−𝑘²𝑆_𝑘⁻²𝛽_LS=𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝛽_LS. (5)

In addition to model (1), let us give some prior information about 𝛽 in the form of a set of 𝐽 which is independent stochastic linear restriction 𝑟=𝑅𝛽+𝑒,𝑒∼0,𝜎²𝑊, (6) where 𝑅 is a 𝐽×𝑝 known matrix of rank 𝐽, 𝑒 is a 𝐽×1 vector of disturbances with expectation 0 and covariance matrix 𝜎²𝑊, 𝑊 is supposed to be known and positive definite, and the 𝐽×1 vector 𝑟 can be interpreted as a random variable with expectation 𝐸(𝑟)=𝑅𝛽. Then, we can derive that (6) does not hold exactly but in the mean. We assume 𝑟 to be a realized value of the random vector, so that all expectations are conditional on 𝑟 [18]. We will not separately mention this in the following discussions. Furthermore, it is also supposed that 𝜀 is stochastically independent of 𝑒.

For the restricted model specified by (1) and (6), the OME introduced by Durbin [10], Theil and Goldberger [11], and Theil [12] is defined as 𝛽_OME=𝑆+𝑅^𝑇𝑊⁻¹𝑅⁻¹𝑋^𝑇𝑦+𝑅^𝑇𝑊⁻¹𝑟. (7)

When the prior information and the sample information are not equally important, Schaffrin and Toutenburg [16] considered the WME which is denoted as 𝛽_WME(𝑤)=𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅⁻¹𝑋^𝑇𝑦+𝑤𝑅^𝑇𝑊⁻¹𝑟, (8) where 𝑤(0≤𝑤≤1) is a nonstochastic and nonnegative scalar weight.

Note that 𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅⁻¹=𝑆⁻¹−𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑅𝑆⁻¹,𝑆⁻¹−𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑅𝑆⁻¹𝑤𝑅^𝑇𝑊⁻¹𝑟=𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑟. (9)(10)

Then, the WME (8) can be rewritten as 𝛽_WME(𝑤)=𝛽_LS+𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑟−𝑅𝛽_LS. (11)

Additionally, by combining the WME and RE, Li and Yang [17] obtained the WMRE which is defined as 𝛽_WMRE()=𝛽𝑤,𝑘_RE(𝑘)+𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑟−𝑅𝛽_RE(𝑘). (12)

The WMRE also can be rewritten as 𝛽_WMRE(𝑤,𝑘)=𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅⁻¹𝑇_𝑘𝑋^𝑇𝑦+𝑤𝑅^𝑇𝑊⁻¹𝑟. (13)

Now, based on the WME [16] and the AURE [9], we can define the following weighted mixed almost unbiased ridge estimator: 𝛽_WMAURE()=𝛽𝑤,𝑘_AURE(𝑘)+𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1×𝑟−𝑅𝛽_AURE(𝑘=)𝐼−𝑘²𝑆_𝑘⁻²𝛽_LS+𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1×𝑟−𝑅𝐼−𝑘²𝑆_𝑘⁻²𝛽_LS=𝑆⁻¹−𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑅𝑆⁻¹×𝐼−𝑘²𝑆_𝑘⁻²𝑋^𝑇𝑦+𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑟, (14) which is according to the way in [17].

Using (10), (14) can be rewritten as 𝛽_WMAURE()=𝑆𝑤,𝑘₋₁−𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑅𝑆⁻¹×𝐼−𝑘²𝑆_𝑘⁻²𝑋^𝑇𝑦+𝑤𝑅^𝑇𝑊⁻¹𝑟=𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅⁻¹𝐼−𝑘²𝑆_𝑘⁻²𝑋^𝑇𝑦+𝑤𝑅^𝑇𝑊⁻¹𝑟. (15)

From the definition of 𝛽_WMAURE(𝑤,𝑘), it can be seen that 𝛽_WMAURE(𝑤,𝑘) is a general estimator, and as special cases of it, the WME, LS, and AURE can be described as 𝛽_WMAURE(𝑤,0)=𝛽_WME(𝑤),𝛽_WMAURE(0,0)=𝛽_LS, (16) and if 𝑅=0, 𝛽_WMAURE(𝑤,𝑘)=𝛽_AURE(𝑘). (17)

It is easy to compute expectation values and covariance matrices of the LS, WME, WMRE, AURE, and WMAURE as 𝐸𝛽_LS𝛽=𝛽,COV_LS=𝜎²𝑆⁻¹,𝐸𝛽_WME(𝑤)𝛽=𝛽,COV_WME(𝑤)=𝜎²𝐴𝑆+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐸𝛽𝐴,_WMRE(𝑤,𝑘)=𝛽+𝐴𝑇_𝑘𝛽−𝐼𝑆𝛽,COV_WMRE(𝑤,𝑘)=𝜎²𝐴𝑇_𝑘𝑆𝑇_𝑘+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐸𝛽𝐴,_AURE(𝑘)=𝐼−𝑘²𝑆_𝑘⁻²𝛽=𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝛽𝛽,COV_AURE(𝑘)=𝜎²𝐼−𝑘²𝑆_𝑘⁻²𝑆⁻¹𝐼−𝑘²𝑆_𝑘⁻²=𝜎²𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆⁻¹𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹,𝐸𝛽_WMAURE(𝑤,𝑘)=𝛽−𝑘²𝐴𝑆_𝑘⁻²𝛽𝑆𝛽,COV_WMAURE(𝑤,𝑘)=𝜎²𝐴𝐼−𝑘²𝑆_𝑘⁻²𝑆𝐼−𝑘²𝑆_𝑘⁻²+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴, (18) where 𝐴=(𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅)⁻¹.

In the rest of the sections, we intend to study the performance of the new estimator over relative estimators under the quadratic bias and the mean square error matrix criteria.

3. Quadratic Bias Comparison of Estimators

In this section, quadratic bias comparisons are performed among the AURE, WMRE, and WMAURE. Let 𝛽 be the estimator of 𝛽, then the quadratic bias of 𝛽 is defined as QB(𝛽)=Bias(𝛽)^𝑇Bias(𝛽), where Bias(𝛽)=𝐸(𝛽)−𝛽. Based on the definition of quadratic bias, we can easily get quadratic biases of AURE, WMRE, and WMAURE: QB𝛽_AURE(𝑘)=Bias𝛽_AURE(𝑘)^𝑇Bias𝛽_AURE(𝑘)=𝑘⁴𝛽^𝑇𝑆_𝑘⁻⁴𝛽𝛽,QB_WMRE()𝛽𝑤,𝑘=Bias_WMRE(𝑤,𝑘)^𝑇Bias𝛽_WMRE(𝑤,𝑘)=𝑘²𝛽^𝑇𝑆𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹𝛽𝑆𝛽,QB_WMAURE()𝛽𝑤,𝑘=Bias_WMAURE(𝑤,𝑘)^𝑇Bias𝛽_WMAURE(𝑤,𝑘)=𝑘⁴𝛽^𝑇𝑆𝑆_𝑘⁻²𝐴²𝑆_𝑘⁻²𝑆𝛽. (19)

3.1. Quadratic Bias Comparison between the AURE and WMAURE

Here, we focus on the quadratic bias comparison between the AURE and WMAURE. The difference of quadratic biases can be derived as Δ₁=QB𝛽_AURE(𝑘)−QB𝛽_WMAURE(𝑤,𝑘)=𝑘⁴𝛽^𝑇𝑆_𝑘⁻⁴𝛽−𝑘⁴𝛽^𝑇𝑆𝑆_𝑘⁻²𝐴²𝑆_𝑘⁻²𝑆𝛽=𝑘⁴𝛽^𝑇𝑆_𝑘⁻⁴𝛽−𝑘⁴𝛽^𝑇𝑆_𝑘⁻²𝑆𝐴²𝑆𝑆_𝑘⁻²𝛽=𝑘⁴𝛽^𝑇𝑆_𝑘⁻²𝐼−𝑆𝐴²𝑆𝑆_𝑘⁻²𝛽. (20)

Firstly, we just consider 𝐼−𝑆𝐴²𝑆. Note that (𝐼−𝑆𝐴²𝑆)^𝑇=𝐼−𝑆𝐴²𝑆 and 𝐼−𝑆𝐴²𝑆=𝑆𝑆⁻²𝑆−𝑆𝐴²𝑆=𝑆𝑆⁻²−𝐴²𝑆. (21)

It can be seen that 𝐴⁻²−𝑆²=(𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅)²−𝑆²>0, namely, 𝐴⁻²>𝑆². Then we can get 𝑆⁻²−𝐴²>0. (22)

Thus, we have 𝐼−𝑆𝐴²𝑆>0 by (21) and (22). Therefore, we can derive that Δ₁=QB(𝛽_AURE(𝑘))−QB(𝛽_WMAURE(𝑤,𝑘))>0 and the 𝛽_WMAURE(𝑤,𝑘) outperforms the 𝛽_AURE(𝑘) according to the quadratic bias criterion.

Based on the above analysis, we can derive the following theorem.

Theorem 1. According to the quadratic bias criterion, the WMAURE performs better than the AURE.

3.2. Quadratic Bias Comparison between the WMRE and WMAURE

Similarly, the quadratic bias comparison between the WMRE and WMAURE will be discussed. The difference of quadratic biases of both estimators can be obtained by Δ₂=QB𝛽_WMRE(𝑤,𝑘)−QB𝛽_WMAURE(𝑤,𝑘)=𝑘²𝛽^𝑇𝑆𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹𝑆𝛽−𝑘⁴𝛽^𝑇𝑆𝑆_𝑘⁻²𝐴²𝑆_𝑘⁻²𝑆𝛽=𝑘²𝛽^𝑇𝑆𝑆_𝑘⁻¹𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹𝑆_𝑘⁻¹𝑆𝛽. (23)

We consider 𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹. Note that (𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹)^𝑇=𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹ and 𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹=𝐴²−𝑘𝑆_𝑘⁻¹𝐴²+𝑘𝑆_𝑘⁻¹𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹=𝐼−𝑘𝑆_𝑘⁻¹𝐴²+𝑘𝑆_𝑘⁻¹𝐴²𝐼−𝑘𝑆_𝑘⁻¹. (24)

For 𝑆=𝑋^𝑇𝑋>0, there exists an orthogonal matrix 𝑃 such that 𝑆=𝑃Λ𝑃^𝑇, where Λ=diag(𝜆₁,…,𝜆_𝑝) and 𝜆₁≥⋯≥𝜆_𝑝 denote the ordered eigenvalues of 𝑆. Therefore, we can easily compute that 𝐼−𝑘𝑆_𝑘⁻¹=𝑃diag(𝜏₁,…,𝜏_𝑝)𝑃^𝑇, where 𝜏_𝑖=1−(𝑘/(𝜆_𝑖+𝑘))=𝜆_𝑖/(𝜆_𝑖+𝑘), 𝑖=1,…,𝑝. For 𝑘>0, 𝜆_𝑖>0, we obtain 𝜏_𝑖>0 which means that 𝐼−𝑘𝑆_𝑘⁻¹>0. Thus, we have (𝐼−𝑘𝑆_𝑘⁻¹)^1/2𝐴²(𝐼−𝑘𝑆_𝑘⁻¹)^1/2>0 and (𝐼−𝑘𝑆_𝑘⁻¹)^1/2𝑘𝑆_𝑘⁻¹𝐴²(𝐼−𝑘𝑆_𝑘⁻¹)^1/2>0. Note that (𝐼−𝑘𝑆_𝑘⁻¹)𝐴², 𝑘𝑆_𝑘⁻¹𝐴²(𝐼−𝑘𝑆_𝑘⁻¹) and (𝐼−𝑘𝑆_𝑘⁻¹)^1/2𝐴²(𝐼−𝑘𝑆_𝑘⁻¹)^1/2, (𝐼−𝑘𝑆_𝑘⁻¹)^1/2𝑘𝑆_𝑘⁻¹𝐴²(𝐼−𝑘𝑆_𝑘⁻¹)^1/2 have same nonzero eigenvalues, respectively, which means that 𝐴²−𝑘²𝑆_𝑘⁻¹𝐴²𝑆_𝑘⁻¹>0.

Therefore, we can derive that Δ₂=QB(𝛽_WMRE(𝑤,𝑘))−QB(𝛽_WMAURE(𝑤,𝑘))>0 and the 𝛽_WMAURE(𝑤,𝑘) performs better than the 𝛽_WMRE(𝑤,𝑘) under the quadratic bias criterion.

We can get the following theorem.

Theorem 2. According to the quadratic bias criterion, the WMAURE outperforms the WMRE.

4. Mean Square Error Matrix Comparisons of Estimators

In this section, We will compare the proposed estimator with relative estimators under the mean square error matrix (MSEM) criterion.

For the sake of convenience, we list some lemmas needed in the following discussions.

Lemma 3. Let 𝑀 be a positive definite matrix, namely, 𝑀>0; let 𝛼 be a vector, then 𝑀−𝛼𝛼^𝑇≥0 if and only if 𝛼^𝑇𝑀⁻¹𝛼≤1.

Proof. See [19].

Lemma 4. Let 𝛽_𝑖=𝐴_𝑖𝑦, 𝑖=1,2 be two competing homogeneous linear estimators of 𝛽. Suppose that 𝐷=𝐶𝑂𝑉(𝛽₁)−𝐶𝑂𝑉(𝛽₂)>0, then 𝑀𝑆𝐸𝑀(𝛽₁)−𝑀𝑆𝐸𝑀(𝛽₂)≥0 if and only if 𝑑^𝑇₂(𝐷+𝑑₁𝑑^𝑇₁)⁻¹𝑑₂≤1, where 𝑀𝑆𝐸𝑀(𝛽_𝑖), 𝑑_𝑖 denote the MSEM and bias vector of 𝛽_𝑖, respectively.

Proof. See [20].

Lemma 5. Let two 𝑛×𝑛 matrices 𝑀>0, 𝑁≥0, then 𝑀>𝑁⇔𝜆₁(𝑁𝑀⁻¹)<1.

Proof. See [18].

Firstly, the MSEM of an estimator 𝛽 is defined as MSEM𝛽=𝐸𝛽−𝛽𝛽−𝛽^𝑇=COV𝛽+Bias𝛽Bias𝛽^𝑇. (25)

For two given estimators 𝛽₁ and 𝛽₂, the estimator 𝛽₂ is said to be superior to 𝛽₁ under the MSEM criterion if and only if 𝑀𝛽₁,𝛽₂=MSEM𝛽₁−MSEM𝛽₂≥0. (26)

The scalar mean square error (MSE) is defined as MSE(𝛽)=tr(MSEM(𝛽)). It is well known that the MSEM criterion is superior over the MSE criterion; we just compare the MSEM of the WMAURE with other relative estimators.

Now, from (18), we can easily obtain the MSEM of the WME, WMRE, AURE, and WMAURE as follows: MSEM𝛽_LS=𝜎²𝑆⁻¹,MSEM𝛽_WME(𝑤)=𝜎²𝐴𝑆+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝛽𝐴,MSEM_WMRE(𝑤,𝑘)=𝜎²𝐴𝑇_𝑘𝑆𝑇_𝑘+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴+𝑏₁𝑏^𝑇₁,MSEM𝛽_AURE(𝑘)=𝜎²𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆⁻¹𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑏₂𝑏^𝑇₂,MSEM𝛽_WMAURE(𝑤,𝑘)=𝜎²𝐴𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑤²𝑅^𝑇𝑊⁻¹𝑅×𝐴+𝑏₃𝑏^𝑇₃, (27)(28)(29)(30)(31) where 𝑏₁=𝐴(𝑇_𝑘−𝐼)𝑆𝛽, 𝑏₂=−𝑘²𝑆_𝑘⁻²𝛽 and 𝑏₃=−𝑘²𝐴𝑆_𝑘⁻²𝑆𝛽.

4.1. MSEM Comparison of the WME and WMAURE

To compare MSEM values between the WMAURE and WME. Firstly, from (28) and (31), the difference of MSEM values between the WME and WMAURE can be gained by 𝑀𝛽_WME(𝑤),𝛽_WMAURE()𝛽𝑤,𝑘=MSEM_WME(𝑤)−MSEM𝛽_WMAURE(𝑤,𝑘)=𝜎²𝐴𝑆+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴−𝜎²𝐴×𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴−𝑏₃𝑏^𝑇₃=𝜎²𝐴𝑆−𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹𝐴−𝑏₃𝑏^𝑇₃=𝜎²𝐷₁−𝑏₃𝑏^𝑇₃, (32) where 𝐷₁=𝐴(𝑆−(𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹))𝐴.

Theorem 6. The WMAURE 𝛽_{𝑊𝑀𝐴𝑈𝑅𝐸}(𝑤,𝑘) is superior to the WME 𝛽_𝑊𝑀𝐸(𝑤) under MSEM criterion, namely, 𝑀(𝛽_𝑊𝑀𝐸(𝑤),𝛽_{𝑊𝑀𝐴𝑈𝑅𝐸}(𝑤,𝑘))≥0 if and only if 𝑏^𝑇₃𝐷₁⁻¹𝑏₃≤𝜎².

Proof. Note that 𝑆=𝑃Λ𝑃^𝑇. We can easily compute that 𝑆−(𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)=𝑃Γ⁽¹⁾𝑃^𝑇=𝑃diag(𝜏₁⁽¹⁾,…,𝜏_𝑝⁽¹⁾)𝑃^𝑇, where Γ⁽¹⁾=Λ−(𝐼+𝑘(Λ+𝑘𝐼)⁻¹)(𝐼+𝑘Λ⁻¹)⁻¹Λ(𝐼+𝑘Λ⁻¹)⁻¹(𝐼+𝑘(Λ+𝑘𝐼)⁻¹) and 𝜏_𝑖⁽¹⁾=𝜆_𝑖−1+𝑘𝜆_𝑖+𝑘⁻¹1+𝑘𝜆_𝑖⁻¹⁻¹×𝜆_𝑖1+𝑘𝜆_𝑖⁻¹⁻¹1+𝑘𝜆_𝑖+𝑘⁻¹=𝜆_𝑖−𝜆_𝑖+2𝑘𝜆_𝑖𝜆+𝑘𝜆^𝑖_𝑖𝜆+𝑘^𝑖𝜆_𝑖𝜆_𝑖𝜆+𝑘^𝑖+2𝑘𝜆_𝑖=𝑘+𝑘₄+4𝑘³𝜆_𝑖+2𝑘²𝜆²_𝑖𝜆_𝑖𝜆_𝑖+𝑘⁴>0,𝑖=1,…,𝑝, (33) which means 𝑆−(𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)>0. Observing that 𝐴=(𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅)⁻¹>0, we have 𝐷₁=𝐴(𝑆−(𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹))𝐴>0.
Applying Lemma 3, we can get that 𝑀(𝛽_WME(𝑤),𝛽_WMAURE(𝑤,𝑘))≥0 if and only if 𝑏^𝑇₃𝐷₁⁻¹𝑏₃≤𝜎².
This completes the proof.

4.2. MSEM Comparison of the WMRE and WMAURE

Similarly, we compare MSEM values between the WMAURE and WMRE. Firstly, from (29) and (31), the difference of MSEM values between the WMRE and the WMAURE can be computed by 𝑀𝛽_WMAURE(𝑤,𝑘),𝛽_WMRE()𝛽𝑤,𝑘=MSEM_WMAURE(𝑤,𝑘)−MSEM𝛽_WMRE(𝑤,𝑘)=𝜎²𝐴𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴+𝑏₃𝑏^𝑇₃−𝜎²𝐴𝑇_𝑘𝑆𝑇_𝑘+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴−𝑏₁𝑏^𝑇₁=𝜎²𝐴𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹−𝑇_𝑘𝑆𝑇_𝑘𝐴−𝑏₁𝑏^𝑇₁=𝜎²𝐷₂+𝑏₃𝑏^𝑇₃−𝑏₁𝑏^𝑇₁, (34) where 𝐷₂=𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)−𝑇_𝑘𝑆𝑇_𝑘)𝐴.

Theorem 7. The WMAURE 𝛽_{𝑊𝑀𝐴𝑈𝑅𝐸}(𝑤,𝑘)(𝑘>0) is superior to the WMRE 𝛽_{𝑊𝑀𝑅𝐸}(𝑤,𝑘) under MSEM criterion, namely, 𝑀(𝛽_{𝑊𝑀𝐴𝑈𝑅𝐸}(𝑤,𝑘),𝛽_{𝑊𝑀𝑅𝐸}(𝑤,𝑘))<0 if and only if 𝑏^𝑇₁(𝜎²𝐷₂+𝑏₃𝑏^𝑇₃)⁻¹𝑏₁>1.

Proof. Firstly, we prove (𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)−𝑇_𝑘𝑆𝑇_𝑘>0.
Note that 𝑆=𝑃Λ𝑃^𝑇, we can compute (𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)−𝑇_𝑘𝑆𝑇_𝑘=𝑃Γ⁽²⁾𝑃^𝑇=𝑃diag(𝜏₁⁽²⁾,…,𝜏_𝑝⁽²⁾)𝑃^𝑇, where Γ⁽²⁾=𝐼+𝑘(Λ+𝑘𝐼)⁻¹1+𝑘Λ⁻¹⁻¹Λ1+𝑘Λ⁻¹⁻¹×𝐼+𝑘(Λ+𝑘𝐼)⁻¹−1+𝑘Λ⁻¹⁻¹Λ1+𝑘Λ⁻¹⁻¹,𝜏_𝑖⁽²⁾=1+𝑘𝜆_𝑖+𝑘⁻¹𝑘1+𝜆_𝑖⁻¹𝜆_𝑖𝑘1+𝜆_𝑖⁻¹×1+𝑘𝜆_𝑖+𝑘⁻¹−𝑘1+𝜆_𝑖⁻¹𝜆_𝑖𝑘1+𝜆_𝑖⁻¹=𝑘1+𝜆_𝑖⁻¹𝜆_𝑖𝑘1+𝜆_𝑖⁻¹𝜆_𝑖+2𝑘𝜆_𝑖+𝑘²>0,𝑖=1,…,𝑝. (35)
Thus, we can get (𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)−𝑇_𝑘𝑆𝑇_𝑘>0. Observing that 𝐴>0, we have 𝐷₂=𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)−𝑇_𝑘𝑆𝑇_𝑘)𝐴>0.
Applying Lemma 4, we can get 𝑀(𝛽_WMAURE(𝑤,𝑘),𝛽_WMRE(𝑤,𝑘))<0 if and only if 𝑏^𝑇₁(𝜎²𝐷₂+𝑏₃𝑏^𝑇₃)⁻¹𝑏₁>1.
This completes the proof.

4.3. MSEM Comparison of the AURE and WMAURE

Now, we compare MSEM values between the WMAURE and AURE. Firstly, from (30) and (31), the difference of MSEM values between the AURE and WMAURE can be obtained by 𝑀𝛽_AURE(𝑘),𝛽_WMAURE()𝛽𝑤,𝑘=MSEM_AURE(𝑘)−MSEM𝛽_WMAURE(𝑤,𝑘)=𝜎²𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑏₂𝑏^𝑇₂−𝑏₃𝑏^𝑇₃−𝜎²𝐴𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴=𝜎²𝐷₃+𝑏₂𝑏^𝑇₂−𝑏₃𝑏^𝑇₃, (36) where 𝐷₃=(𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)−𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴.

Theorem 8. When 𝜆₁(𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹))⁻¹)<1, the WMAURE is superior to the AURE in the MSEM sense, namely, 𝑀(𝛽_{𝐴𝑈𝑅𝐸}(𝑘),𝛽_{𝑊𝑀𝐴𝑈𝑅𝐸}(𝑤,𝑘))≥0 if and only if 𝑏^𝑇₃(𝜎²𝐷₃+𝑏₂𝑏^𝑇₂)⁻¹𝑏₃≤1.

Proof. Note that (𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)>0 and 𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴>0. When 𝜆₁(𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹))⁻¹)<1, we can get that 𝐷₃>0 by applying Lemma 5. Thus, from (36) and applying Lemma 4, we have 𝑀(𝛽_AURE(𝑘),𝛽_WMAURE(𝑤,𝑘))≥0 if and only if 𝑏^𝑇₃(𝜎²𝐷₃+𝑏₂𝑏^𝑇₂)⁻¹𝑏₃≤1.
The proof is completed.

4.4. MSEM Comparison of the LS and WMAURE

Finally, we compare MSEM values between the LS and WMAURE. From (27) and (31), the difference of MSEM values between the LS and WMAURE can be computed by 𝑀𝛽_LS,𝛽_WMAURE()𝛽𝑤,𝑘=MSEM_LS−MSEM𝛽_WMAURE(𝑤,𝑘)=𝜎²𝑆⁻¹−𝜎²𝐴𝐼+𝑘𝑆_𝑘⁻¹𝑇_𝑘𝑆𝑇_𝑘𝐼+𝑘𝑆_𝑘⁻¹+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴−𝑏₃𝑏^𝑇₃=𝜎²𝑆⁻¹−𝜎²𝐴+𝜎²𝐴−MSEM𝛽_WME(𝑤)+MSEM𝛽_WME(𝑤)−MSEM𝛽_WMAURE()=𝜎𝑤,𝑘₂𝑆⁻¹−𝜎²𝐴+𝜎²𝐴−𝜎²𝐴𝑆+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴+𝜎²𝐷₁−𝑏₃𝑏^𝑇₃, (37) where 𝐴=(𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅)⁻¹ and 𝐷₁=𝐴(𝑆−(𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹))𝐴>0 according to Section 4.1.

Firstly, using (9), we can compute that 𝜎²𝑆⁻¹−𝜎²𝐴=𝜎²𝑤𝑆⁻¹𝑅^𝑇𝑊+𝑤𝑅𝑆⁻¹𝑅^𝑇−1𝑅𝑆⁻¹>0. (38)

Moreover, it can be computed that 𝜎²𝐴−𝜎²𝐴𝑆+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴=𝜎²𝐴𝐴⁻¹𝐴−𝜎²𝐴𝑆+𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴=𝜎²𝐴𝑆+𝑤𝑅^𝑇𝑊⁻¹𝑅−𝑆−𝑤²𝑅^𝑇𝑊⁻¹𝑅𝐴=𝑤(1−𝑤)𝜎²𝐴𝑅^𝑇𝑊⁻¹𝑅𝐴>0. (39)

Therefore, 𝜎²𝑆⁻¹−𝜎²𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴>0. Applying Lemma 3, we can get that 𝑀(𝛽_LS,𝛽_WMAURE(𝑤,𝑘))≥0 if and only if 𝑏^𝑇₃(𝜎²𝑆⁻¹−𝜎²𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴)𝑏₃≤0.

Based on the above analysis, we can state the following theorem.

Theorem 9. The WMAURE is superior to the LS according to the MSEM criterion, namely, 𝑀(𝛽_𝐿𝑆,𝛽_{𝑊𝑀𝐴𝑈𝑅𝐸}(𝑤,𝑘))≥0 if and only if 𝑏^𝑇₃(𝜎²𝑆⁻¹−𝜎²𝐴((𝐼+𝑘𝑆_𝑘⁻¹)𝑇_𝑘𝑆𝑇_𝑘(𝐼+𝑘𝑆_𝑘⁻¹)+𝑤²𝑅^𝑇𝑊⁻¹𝑅)𝐴)𝑏₃≤0.

5. Selection of Parameters 𝑘 and 𝑤

In this section, we give a method about how to choose parameters 𝑘 and 𝑤. Firstly, a linear regression model can be transformed to a canonical form by the orthogonal transformation. Let 𝑄 be an orthogonal matrix such that 𝑄^𝑇𝑋^𝑇𝑋𝑄=Λ=diag(𝜆₁,…,𝜆_𝑝), where 𝜆_𝑖 is the eigenvalue of 𝑋^𝑇𝑋, and 𝑋=𝑋𝑄, 𝛽=𝑄^𝑇𝛽=(𝛽₁,…,𝛽_𝑝)^𝑇. Then, we get a canonical form of model (1) as 𝑦=𝑋𝛽+𝜀. (40)

Note that 𝛽_WMAURE(𝑤,𝑘)=𝑄^𝑇𝛽_WMAURE(𝑤,𝑘) and MSEM(𝛽_WMAURE(𝑤,𝑘))=𝑄^𝑇MSEM(𝛽_WMAURE(𝑤,𝑘))𝑄. It is supposed that 𝑆 and 𝑅^𝑇𝑊⁻¹𝑅 are commutative, then we have MSEM𝛽_WMAURE(𝑤,𝑘)=𝜎²(Λ+𝑤Ψ)⁻¹𝐼+𝑘(Λ+𝑘𝐼)⁻¹(Λ+𝑘𝐼)⁻¹×Λ³(Λ+𝑘𝐼)⁻¹𝐼+𝑘(Λ+𝑘𝐼)⁻¹+𝑤²Ψ×(Λ+𝑤Ψ)⁻¹+𝑘⁴(Λ+𝑤Ψ)⁻¹(Λ+𝑘𝐼)⁻²×Λ𝛽𝛽^𝑇Λ(Λ+𝑘𝐼)⁻²(Λ+𝑤Ψ)⁻¹, (41) where 𝑄^𝑇𝑅^𝑇𝑊⁻¹𝑅𝑄=Ψ=diag(𝜉₁,…,𝜉_𝑝).

Optimal values for 𝑘 and 𝑤 can be derived by minimizing 𝑔(𝑤,𝑘)=trMSEM𝛽_WMAURE()=𝑤,𝑘_𝑝𝑖=1𝜎²𝜆³_𝑖𝜆_𝑖+2𝑘²+𝑤𝜉²_𝑖𝜆_𝑖+𝑘⁴+𝑘⁴𝜆²_𝑖𝛽²_𝑖𝜆_𝑖+𝑤𝜉_𝑖²𝜆_𝑖+𝑘⁴. (42)

For a fixed value of 𝑤, differentiating 𝑔(𝑤,𝑘) with respect to 𝑘 leads to 𝜕𝑔(𝑤,𝑘)=𝜕𝑘_𝑝𝑖=1𝜎²4𝜆_𝑖+2𝑘𝜆³_𝑖+4𝑤𝜉²_𝑖𝜆_𝑖+𝑘³+4𝑘³𝜆²_𝑖𝛽²_𝑖×𝜆_𝑖+𝑘−4𝜎²𝜆_𝑖+2𝑘²𝜆³_𝑖+𝑤𝜉²_𝑖𝜆_𝑖+𝑘⁴+𝑘⁴𝜆²_𝑖𝛽²_𝑖𝜆_𝑖+𝑤𝜉_𝑖²𝜆_𝑖+𝑘⁵⁻¹=^𝑝_𝑖=14𝑘𝜆³_𝑖𝑘²𝛽²_𝑖−2𝜎²𝑘−𝜎²𝜆_𝑖𝜆_𝑖+𝑤𝜉_𝑖²𝜆_𝑖+𝑘⁵, (43) and equating it to zero. Note that 𝑘>0 and after unknown parameters 𝜎² and 𝛽 are replaced by their unbiased estimators, we obtain the optimal estimator of 𝑘 for a fixed 𝑤 value as 𝜎𝑘=²∑^𝑝_𝑖=1𝜆³_𝑖+𝜎⁴∑^𝑝_𝑖=1𝜆³_𝑖²+𝜎²∑^𝑝_𝑖=1𝜆⁴_𝑖∑^𝑝_𝑖=1𝜆³_𝑖𝛽²_𝑖∑^𝑝_𝑖=1𝜆³_𝑖𝛽²_𝑖. (44)

The 𝑤 value which minimizes the function 𝑔(𝑤,𝑘) can be found by differentiating 𝑔(𝑤,𝑘) with respect to 𝑤 when 𝑘 is fixed 𝜕𝑔(𝑤,𝑘)=𝜕𝑤_𝑝𝑖=1𝜎²𝜉²_𝑖𝜆_𝑖+𝑘⁴𝜆_𝑖+𝑤𝜉_𝑖−2𝑤𝜎²𝜆³_𝑖𝜆_𝑖+2𝑘²+𝑤𝜎²𝜉²_𝑖𝜆_𝑖+𝑘⁴+𝑘⁴𝜆²_𝑖𝛽²_𝑖×𝜆_𝑖+𝑤𝜉_𝑖³𝜆_𝑖+𝑘⁴⁻¹=^𝑝_𝑖=1−2𝑤²𝜎²𝜉²_𝑖𝜆_𝑖+𝑘⁴+𝜎²𝜆_𝑖𝜉²_𝑖𝜆_𝑖+𝑘⁴+𝑤𝜎²𝜉³_𝑖𝜆_𝑖+𝑘⁴−2𝜎²𝜆³_𝑖𝜆_𝑖+2𝑘²−2𝑘⁴𝜆²_𝑖𝛽²_𝑖×𝜆_𝑖+𝑤𝜉_𝑖³𝜆_𝑖+𝑘⁴⁻¹ (45) and equating it to zero. After unknown parameters 𝜎² and 𝛽 are replaced by their unbiased estimators, we get the optimal estimator of 𝑤 for a fixed 𝑘 value as 𝑤=∑^𝑝_𝑖=1ℎ²_2𝑖+8∑^𝑝_𝑖=1ℎ_1𝑖²−∑^𝑝_𝑖=1ℎ_2𝑖∑^𝑝_𝑖=14ℎ_1𝑖, (46) where ℎ_1𝑖=𝜎²𝜉²_𝑖(𝜆_𝑖+𝑘)⁴ and ℎ_2𝑖=2𝜎²𝜆³_𝑖(𝜆_𝑖+2𝑘)²+2𝑘⁴𝜆²_𝑖𝛽²_𝑖−𝜎²𝜉³_𝑖(𝜆_𝑖+𝑘)⁴.

6. Numerical Example and Monte Carlo Simulation

In order to verify our theoretical results, we firstly conduct an experiment based on a real data set originally due to Woods et al. [21]. In this experiment, we replace the unknown parameters 𝛽 and 𝜎² by their unbiased estimators, which is according to the way in [17]. The result here and below is performed with R 2.14.1.

We can easily obtain that the condition number is about 3.66×10⁷. This information indicates a serious multicollinearity among the regression vector. The ordinary least squares estimator of 𝛽 is 𝛽_LS=𝑆⁻¹𝑋^𝑇𝑦=(62.4054,1.5511,0.5102,0.1019,−0.1441)^𝑇 (47) with 𝜎²_LS=5.983.

Consider the following stochastic linear restrictions used in [17]: ,𝜎𝑟=𝑅𝛽+𝑒,𝑅=1111101311𝑟=63.94982.5648,𝑒∼0,²_LS𝐼₂. (48)

For the WMRE, AURE, and WMAURE, their quadratic bias values are given in Table 1 and their estimated MSE values are obtained in Table 2 by replacing all unknown parameters in the corresponding theoretical MSE expressions by their least squares estimators.

Table 1 

Estimated QB values of the WMRE, AURE, and WMAURE.

Table 2 

Estimated MSE values of the WME, WMRE, AURE, and WMAURE.

It can be seen from Table 1 that the WMAURE has smaller quadratic bias values than the WMRE and AURE for every case, which agrees with our theoretical finding in Section 3. From Table 2, we can get that MSE values of our proposed estimator are the smallest among the LS, WME, WMRE, AURE, and WMAURE when 𝑤 is fixed, which agrees with our theoretical finding in Theorems 6–9.

To further illustrate the behavior of our proposed estimator, we are to perform a Monte Carlo simulation study under different levels of multicollinearity. Following the way in [22, 23], we can get explanatory variables by the following equations: 𝑥_𝑖𝑗=1−𝜌^21/2𝑧_𝑖𝑗+𝜌𝑧_𝑖,𝑝+1,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑝, (49) where 𝑧_𝑖𝑗 is an independent standard normal pseudorandom number, and 𝜌 is specified so that the theoretical correlation between any two explanatory variables is given by 𝜌². A dependent variable is generated by 𝑦_𝑖=𝛽₁𝑥_𝑖1+𝛽₂𝑥_𝑖2+𝛽₃𝑥_𝑖3+𝛽₄𝑥_𝑖4+𝜀_𝑖,𝑖=1,2,…,𝑛, (50) where 𝜀_𝑖 is a normal pseudo-random number with mean zero and variance 𝜎²_𝑖. In this study, we choose (𝛽₁,𝛽₂,𝛽₃,𝛽₄)^𝑇=(40,1,2,3)^𝑇, 𝑛=60, 𝑝=4, 𝜎²_𝑖=1, and the stochastic restriction 𝑟=𝑅𝛽+𝑒, 𝑅=⁴⁰⁻³¹₂₁₂₀, 𝑒∼𝑁(0,0.1𝐼₂). Furthermore, we discuss three cases when 𝜌=0.8,0.9,0.99.

For three different levels of multicollinearity, MSE values of LS, WME, AURE, WMRE, and WMAURE are obtained in Tables 3, 4, and 5, respectively. From Tables 3–5, we can derive the following results.(1)With the increase of multicollinearity, MSE values of the LS, WME, WMRE, AURE, and WMAURE are increasing. And for all cases, the WMAURE has smaller estimated MSE values than the LS, AURE, and WME. (2)The value of 𝑤 is the level of the weight to the sample information and the prior information; we can see from three tables that estimated MSE values of the WME, WMRE, and WMAURE become more and more smaller when the value of 𝑤 increases. It can be concluded that we get more exact estimator of the parameter with more depended prior information.

Table 3 

Estimated MSE values of the WME, WMRE, AURE, and WMAURE when 𝜌=0.8.

Table 4 

Estimated MSE values of the WME, WMRE, AURE, and WMAURE when 𝜌=0.9.

Table 5 

Estimated MSE values of the WME, WMRE, AURE, and WMAURE when 𝜌=0.99.

7. Conclusions

In this paper, we propose the WMAURE based on the WME [16] and the AURE [9] and discuss some properties of the new estimator over the relative estimators. In particular, we prove that the WMAURE has smaller quadratic bias than the AURE and WMRE and derive that the proposed estimator is superior to the LS, WME, WMRE, and AURE in the mean squared error matrix sense under certain conditions. The optimal values of parameters 𝑘 and 𝑤 are obtained. Furthermore, we perform a real data example and a Monte Carlo study to support the finding of our theoretical results.

Acknowledgments

The authors would like to thank the editor and the two anonymous referees for their very helpful suggestions and comments, which improve the presentation of the paper. Also, the authors acknowledge the financial support of the National Natural Science Foundation of China, project no. 11171361, and Ph.D. Programs Foundation of Ministry of Education of China, project no. 20110191110033.