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

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.

It is well known that the ordinary least squares estimator (LS) for is given bŷL 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], − class estimator [3], Liu estimator (LE) [4], and − 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 and 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.

The Proposed Estimator
The ordinary ridge estimator proposed by Hoerl and Kennard [2] is defined aŝ The almost unbiased ridge estimator obtained by Akdeniz and Erol [9] is denoted aŝ In addition to model (1), let us give some prior information about in the form of a set of which is independent stochastic linear restriction where is a × known matrix of rank , is a × 1 vector of disturbances with expectation 0 and covariance matrix 2 , is supposed to be known and positive definite, and the × 1 vector can be interpreted as a random variable with expectation ( ) = . Then, we can derive that (6) does not hold exactly but in the mean. We assume to be a realized value of the random vector, so that all expectations are conditional on [18]. We will not separately mention this in the following discussions. Furthermore, it is also supposed that is stochastically independent of . For the restricted model specified by (1) and (6), the OME introduced by Durbin [10], Theil and Goldberger [11], and Theil [12] is defined aŝ When the prior information and the sample information are not equally important, Schaffrin and Toutenburg [16] considered the WME which is denoted aŝ where (0 ≤ ≤ 1) is a nonstochastic and nonnegative scalar weight.
Note that Then, the WME (8) can be rewritten aŝ Additionally, by combining the WME and RE, Li and Yang [17] obtained the WMRE which is defined aŝ The WMRE also can be rewritten aŝ Now, based on the WME [16] and the AURE [9], we can define the following weighted mixed almost unbiased ridge estimator: which is according to the way in [17].
Journal of Applied Mathematics 3 Using (10), (14) can be rewritten aŝ From the definition of̂W MAURE ( , ), it can be seen that WMAURE ( , ) is a general estimator, and as special cases of it, the WME, LS, and AURE can be described aŝ WMAURE ( , 0) =̂W ME ( ) , It is easy to compute expectation values and covariance matrices of the LS, WME, WMRE, AURE, and WMAURE as 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.

Quadratic Bias Comparison of Estimators
In this section, quadratic bias comparisons are performed among the AURE, WMRE, and WMAURE. Let̂be the estimator of , then the quadratic bias of̂is defined as QB(̂) = Bias(̂) Bias(̂), where Bias(̂) = (̂) − . Based on the definition of quadratic bias, we can easily get quadratic biases of AURE, WMRE, and WMAURE:
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.
We can get the following theorem.

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

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. Proof. See [19]. Proof. See [20].

Selection of Parameters and
In this section, we give a method about how to choose parameters and .  For a fixed value of , differentiating ( , ) with respect to leads to ( , ) and equating it to zero. Note that > 0 and after unknown parameters 2 and̃are replaced by their unbiased estimators, we obtain the optimal estimator of for a fixed value aŝ The value which minimizes the function ( , ) can be found by differentiating ( , ) with respect to when is fixed and equating it to zero. After unknown parameters 2 andã re replaced by their unbiased estimators, we get the optimal estimator of for a fixed value aŝ

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 and 2 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 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. 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 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: where is an independent standard normal pseudorandom number, and is specified so that the theoretical correlation between any two explanatory variables is given by 2 . A dependent variable is generated by where is a normal pseudo-random number with mean zero and variance 2 . In this study, we choose ( 1 , 2 , 3 , 4 ) = (40, 1, 2, 3) , = 60, = 4, 2 = 1, and the stochastic restriction = + , = ( 4 0 −3 1 2 1 2 0 ), ∼ (0, 0.1 2 ). Furthermore, we discuss three cases when = 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 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 increases. It can be concluded that we get more exact estimator of the parameter with more depended prior information.

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 and are obtained. Furthermore, we perform a real data example and a Monte Carlo study to support the finding of our theoretical results.