# A Modified New Two-Parameter Estimator in a Linear Regression Model

Authors: Adewale F. Lukman; Kayode Ayinde; Sek Siok Kun; Emmanuel T. Adewuyi
Journal: Modelling and Simulation in Engineering
 (2019)

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## Abstract

The literature has shown that ordinary least squares estimator (OLSE) is not best when the explanatory variables are related, that is, when multicollinearity is present. This estimator becomes unstable and gives a misleading conclusion. In this study, a modified new two-parameter estimator based on prior information for the vector of parameters is proposed to circumvent the problem of multicollinearity. This new estimator includes the special cases of the ordinary least squares estimator (OLSE), the ridge estimator (RRE), the Liu estimator (LE), the modified ridge estimator (MRE), and the modified Liu estimator (MLE). Furthermore, the superiority of the new estimator over OLSE, RRE, LE, MRE, MLE, and the two-parameter estimator proposed by Ozkale and Kaciranlar (2007) was obtained by using the mean squared error matrix criterion. In conclusion, a numerical example and a simulation study were conducted to illustrate the theoretical results.


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## Body

## 1. Introduction

The general linear regression model in matrix form is defined as(1)y=Xβ+ε,where y is a n×1 vector of the dependent variable, X is a known n×p full-rank matrix of explanatory variables, β is a p×1 vector of regression coefficients, and ε is n×1 vector of disturbance such that Eε=0 and Covε=σ2I. The ordinary least squares estimator (OLSE) of β in model (1) is defined as(2)β^OLS=X′X−1X′y.According to the Gauss–Markov theorem, the OLS estimator is considered best, linear, and unbiased, possessing minimum variance in the class of all linear unbiased estimators. However, different studies have shown that the OLS estimator is not best when the explanatory variables are related, that is, when multicollinearity is present [1]. This estimator becomes unstable and gives a misleading conclusion. Many biased estimators have been proposed as an alternative to OLSE to circumvent this problem. These include Stein estimator [2], principal components estimator [3], ridge estimator (RRE) estimator [1], contraction estimator [4], modified ridge regression estimator (MRRE) [5], and Liu estimator [6].Hoerl and Kennard [1] proposed a ridge estimator (RRE)(3)β^RREk=X′X+kI−1X′Xβ^OLSy=Tkβ^OLS,k>0,where Tk=X′X+kI−1X′X. β^RREk was obtained by augmenting the equation 0=k1/2β+ε′ to the original equation (1) and then applying the OLS estimator. Mayer and Willke [4] defined the contraction estimator(4)β^ρ=1+ρ−1β^,ρ>0.Liu [6] combined the Stein estimator with a ridge estimator to combat the problem of multicollinearity. β^LEd was obtained by augmenting the equation dβ^=β+ε′ to the original equation (1) and then applying OLS. This is defined as follows:(5)β^LEd=X′X+I−1X′y+dIβ^OLS=Tdβ^OLS,0<d<1,where Td=X′X+I−1X′y+dI.Swindel [5] modified the ridge estimator (MRRE) by adding a prior information. The estimator is defined as follows:(6)β^MRREk,b=X′X+kI−1X′y+kb,where b represent the prior information on β. MRRE tends to b as k tends to infinity. Also, MRRE returns the estimates of the OLS estimator when k = 0.Based on prior information, Li and Yang [7] proposed a modified Liu estimator (MLE):(7)β^MLEd,b=X′X+I−1X′X+dIβ^OLS+1−db.MLE includes OLS and Liu as special cases. In recent times, different researchers have suggested the use of two-parameter estimators to handle multicollinearity. Ozkale and Kaciranlar [8] proposed the two-parameter estimator (TPE), which is defined as(8)β^TPEk,d=X′X+kI−1X′Y+kdβ^OLS=X′X+kI−1X′X+kdβ^OLS=Tkdβ^,where k>0,0<d<1. TPE includes OLS, RRE, LE, and the contraction estimators as special cases.The primary focus of this study is to provide an alternative method in a linear regression model to circumvent the problem of multicollinearity. A modified two-parameter (MTP) estimator is proposed based on prior information and is compared with OLS, LE, RRE, MRRE, MLE, and TPE, respectively, using the mean squared error matrix (MSEM) criterion. The article is structured as follows: We introduce the new estimator in Section2. In Section 3, we discuss the superiority of the new estimator. Section 4 consists of a numerical example and a simulation study. Concluding remarks are provided in Section 5.


## 2. Modified Two-Parameter Estimator

LetTk=X′X+kI−1X′X=I−kX′X+kI−1, and MRRE in equation (6) can be re-expressed as(9)β^MRREk,b=X′X+kI−1X′y+kX′X+kI−1b=X′X+kI−1X′Xβ^OLS+kX′X+kI−1b=Tkβ^OLS+I−Tkb.Similarly,Td=X′X+I−1X′X+dI, and then the modified Liu estimator in equation (7) can be written as(10)β^MLEd,b=Tdβ^OLS+I−Tdb=X′X+I−1X′X+dIβ^OLS+1−db.MRRE and MLE are the convex combination of the prior informationb and the OLS estimator. From equation (8), Tkd=X′X+kI−1X′X+kdI=I−k1−dX′X+kI−1; therefore, the modified two-parameter based on the prior information can be defined as follows:(11)β^MTPEk,d,b=Tkdβ^OLS−I−Tkdb=X′X+kI−1X′X+kdIβ^OLS+I−X′X+kI−1X′X+kdIb=X′X+kI−1X′X+kdIβ^OLS+k1−dX′X+kI−1b=X′X+kI−1X′X+kdIβ^OLS+k1−db.Also, MTPE is a convex combination of the prior information and OLSE. It includes the special cases of OLSE, RRE, MRE, LE, and MLE. The following cases are possible:β^MTPEk,1,b0=β^MTPE0,d,b0=β^OLS; ordinary least squares estimatorβ^MTPE1,d,0=β^LEd; Liu estimatorβ^MTPEk,0,b0=β^MRREk,b0; modified ridge estimatorβ^MTPEk,0,0=β^RREk; ridge estimatorβ^MTPE1,d,b0=β^MLEd,b0; modified Liu estimatorSuppose there exist an orthogonal matrixT such that T′X′XT=Λ=diagλ1,λ2,…,λp, where λi is the ith eigenvalue of X′X. Λ and T are the matrices of eigenvalues and eigenvectors of X′X, respectively. Substituting Z=XQ, α=Q′β in model (1), then the equivalent model can be rewritten as(12)y=Zα+ε.The following representations of the estimators are as follows:(13)α^OLS=Λ−1Z′Y,α^LEd=Λ+I−1Λ+dIΛ−1Z′y,α^RREk=Λ+kI−1Z′y,α^MRREk,b=Λ+kI−1Z′y+kb,α^MLEd,b=Λ+I−1Λ+dIΛ−1Z′y+1−db,α^TPEk,d=Λ+kI−1Λ+kdIΛ−1Z′y,α^MTPEk,d,b=Λ+kI−1Λ+kdIΛ−1Z′y+k1−db.The following notations and lemmas are needful to prove the statistical property ofβ^MTPEk,d,b0.Lemma 1.
LetM be an n×n positive definite matrix, that is, M > 0, and α be some vector, then M−αα′≥0 if and only if α′M−1α≤1 [9].Lemma 2.
Letβ^i=Aiy,i=1,2 be two linear estimators of β. Suppose that D=Covβ^1−Covβ^2>0, where Covβ^i,i=1,2 denotes the covariance matrix of β^i and bi=Biasβ^i=AiX−Iβ,i=1,2. Consequently,(14)Δβ^1−β^2=MSEMβ^1−MSEMβ^2=σ2D+b1b1′−b2b2′>0,if and only if b2′σ2D+b1b1′−1b2<1, where MSEMβ^i=Covβ^i+bibi′ [10].


## 3. Establishing Superiority of Modified Two-Parameter Estimator Using MSEM Criterion

In this section, MTPE is compared with the following estimators: OLS, RRE, LE, MRRE, MLE, and TPE.


### 3.1. Comparison between the MTPE and OLS Using MSEM Criterion

From the representationα^MTPEk,d,b=Λ+kI−1Λ+kdIΛ−1Z′y+k1−db, the bias vector and covariance matrix of MTPE are obtained as follows:(15)Eα^MTPEk,d,b=EΛ+kI−1Λ+kdIΛ−1Z′y+k1−db=Λ+kI−1Λ+kdIΛ−1Z′Zα+k1−db=Λ+kI−1Λ+kdIα+k1−db=Λ+kI−1Λ+kdIα+Λ+kI−1k1−db,where Ey=Zα.Recall thatk1−d=Λ+kI−Λ+kdI and let Bk,d,b=Λ+kI−1Λ+kdI. Therefore,(16)Eα^MTPEk,d,b=Bk,d,bα+I−Bk,d,bb,biasα^MTPEk,d,b=Bk,d,bα+I−Bk,d,bb−α,=Bk,d,bα−Iα−b,(17)Covα^MTPEk,d,b=σ2Bk,d,bΛ−1Bk,d,b′.Hence,(18)MSEMα^MTPEk,d,b=σ2Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iα−bα−b′Bk,d,b−I′.From the representation,α^=Λ−1Z′Y, the MSEM of OLS is given as(19)MSEMα^OLS=σ2Λ−1.Comparing (18) and (19),(20)MSEMα^OLS−MSEMα^MTPEk,d,b=σ2Λ−1−Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iα−bα−b′Bk,d,b−I′.Letk > 0 and 0 < d < 1. Thus, the following theorem holds.Theorem 3.
Consider two biased competing homogenous linear estimatorsα^OLS and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to estimator α^ using the MSEM criterion, that is, MSEMα^OLS−MSEMα^MTPEk,d,b>0 if and only if(21)α−b′Bk,d,b−I′σ2Λ−1−Bk,d,bΛ−1Bk,d,b′−1Bk,d,b−Iα−b<1.Proof.
Using (17) and (19), the following was obtained:(22)Covα^−Covα^MTPEk,d,b=σ2Λ−1−Bk,d,bΛ−1Bk,d,b′=σ2diag1λi−λi+kd2λiλi+k2i=1p.Λ−1−Bk,d,bΛ−1Bk,d,b′ will be positive definite (pd) if and only if λi+k2−λi+kd2>0 or λi+k−λi+kd>0. It was observed that λi+k−λi+kd=k1−d>0 for 0<d<1 and k > 0. Therefore, Λ−1−Bk,d,bΛ−1Bk,d,b′ is pd. By Lemma 2, the proof is completed.


### 3.2. Comparison between the MTPE and RRE Using MSEM Criterion

From the representation,α^RREk=Λ+kI−1Z′y, the bias vector and covariance matrix of RRE is given as follows:(23)biasα^RREk=−kΛ+kI−1α,Covα^RREk=σ2Λ+kI−1ΛΛ+kI−1.Hence,(24)MSEMα^RREk=σ2BkΛBk′+k2Bkαα′Bk′,where Bk=Λ+kI−1. The difference between α^RREk and α^MTPEk,d,b in the MSEM sense is as follows:(25)MSEMα^RREk−MSEMα^MTPEk,d,b=σ2BkΛBk′−Bk,d,bΛ−1Bk,d,b′+k2Bkαα′Bk−Bk,d,b−Iα−bα−b′Bk,d,b−I′.Letk > 0 and 0 < d < 1. Thus, the following theorem holds.Theorem 4.
Consider two biased competing homogenous linear estimatorsα^RREk and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to estimator α^RREk using the MSEM criterion, that is, MSEMα^RREk−MSEMα^MTPEk,d,b>0 if and only if(26)α−b′Bk,d,b−I′σ2BkΛBk′−Bk,d,bΛ−1Bk,d,b′+k2Bkαα′Bk−1Bk,d,b−Iα−b<1.


### 3.3. Comparison between the MTPE and LE Using MSEM Criterion

From the representation,α^LEd=Λ+I−1Λ+dIΛ−1Z′y, the bias vector and covariance matrix of RRE are provided as follows:(27)biasα^LEd=Bd−Iα,(28)Covα^LEd=σ2BdΛ−1Bd′.Hence,(29)MSEMα^LEd=σ2BdΛ−1Bd′+Bd−Iαα′Bd−I′,where Bd=Λ+I−1Λ+dI. Considering the difference between (18) and (29),(30)MSEMα^LEd−MSEMα^MTPEk,d,b=σ2D+b1b1′−b2b2′,where D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′, b1=Bd−Iα, and b2=Bk,d,b−Iα−b.Theorem 5.
Consider two biased competing homogenous linear estimatorsα^LEd and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to estimator α^LEd using the MSEM criterion, that is, MSEMα^LEd−MSEMα^MTPEk,d,b>0 if and only if(31)α−b′Bk,d,b−I′σ2BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′+Bd−Iαα′Bd−I′−1Bk,d,b−Iα−b<1.Proof.
Using (17) and (28), the following was obtained:
By computation,(32)σ2D=σ2BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′=σ2diagλi+d2λiλi+12−λi+kd2λiλi+k2.
B
d
Λ
−
1
B
d
′
−
B
k
,
d
,
b
Λ
−
1
B
k
,
d
,
b
′ will be positive definite (pd) if and only if, λi+d2λi+k2−λi+12λi+kd2>0. For 0<d<1 and k > 1, it was observed that λi+d2λi+k2−λi+12λi+kd2>0. Therefore, BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′ is pd. By Lemma 2, the proof is completed.


### 3.4. Comparison between the MTPE and MRRE Using MSEM Criterion

From the representation,α^MRREk,b=Λ+kI−1Z′y+kb, the bias vector and covariance matrix of MRRE are provided as follows:(33)biasα^MRREk,b=Bk−Iα−b,(34)Covα^MRREk,b=σ2BkΛ−1Bk′.Hence,(35)MSEMα^MRREk,b=σ2BkΛ−1Bk′+Bk−Iα−bα−b′Bk−I′,where Bk=Λ+kI−1. Considering the difference between (18) and (35),(36)MSEMα^MRREk−MSEMα^MTPEk,d,b=σ2BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′+Bk−Iα−bα−b′Bk−I′−Bk,d,b−Iα−bα−b′Bk,d,b−I′.Theorem 6.
Consider two biased competing homogenous linear estimatorsα^MRREk,b and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to the estimator α^MRREk,b using the MSEM criterion, that is, MSEMα^MRREk,b−MSEMα^MTPEk,d,b>0 if and only if(37)Bk,d,b−I′α−b′σ2BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′+Bk−Iα−bα−b′Bk−I′−1Bk,d,b−Iα−b<1.Proof.
Using (17) and (34), the following was obtained:(38)σ2BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′=σ2diagλiλi+k2−λi+kd2λiλi+k2.
Evidently, for0<d<1 and k > 0, BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′ will be positive definite (pd). By Lemma 2, the proof is completed.


### 3.5. Comparison between the MTPE and MLE Using MSEM Criterion

From the representation,α^MLEd,b=Λ+I−1Λ+dIΛ−1Z′y+1−db, the bias vector and covariance matrix of MLE are provided as follows:(39)biasα^MLEd=Bd−Iα−b,(40)Covα^MLEd=σ2BdΛ−1Bd′.Hence,(41)MSEMα^MLEd=σ2BdΛ−1Bd′+Bd−Iα−bα−b′Bd−I′,where Bd=Λ+I−1Λ+dI. The mean square error difference between (18) and (41) is given as(42)Δ1=MSEMα^MLEd−MSEMα^MTPEk,d,b=σ2D+b1b1′−b2b2′,where D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′, b1=Bd−Iα−b, b2=Bk,d,b−Iα−b.Theorem 7.
Consider two biased competing homogenous linear estimatorsα^MLEd,b and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to the estimator α^MLEd,b using the MSEM criterion, that is, MSEMα^MLEd,b−MSEMα^MTPEk,d,b>0 if and only if b2′σ2D+b1b1′−1b2<1, where D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′, b1=Bd−Iα−b, and b2=Bk,d,b−Iα−b.Proof.
Using (17) and (40), the following was obtained:
By computation,(43)D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′=Qdiagτ1,…,τpQ′.
By computation,(44)σ2D=σ2BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′=σ2diagλi+d2λiλi+12−λi+kd2λiλi+k2.
σ
2
B
d
Λ
−
1
B
d
′
−
B
k
,
d
,
b
Λ
−
1
B
k
,
d
,
b
′ will be positive definite if and only if λi+d2λi+k2−λi+kd2λi+12>0.


### 3.6. Comparison between the MTPE and TPE Using MSEM Criterion

From the representationα^TPEk,d=Λ+kI−1Λ+kdIΛ−1Z′y, the bias vector and covariance matrix of TPE are provided as follows:(45)biasα^TPEk,d=Bk,d,bα−Iα,(46)Covα^TPEk,d=σ2Bk,d,bΛ−1Bk,d,b′.Hence,(47)MSEMα^TPEk,d=σ2Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iαα′Bk,d,b−I′.Considering the matrix difference between (18) and (47)(48)Δ2=MSEMα^TPEk,d−MSEMα^MTPEk,d,b=Bk,d,b−Iαα′−α−bα−b′Bk,d,b−I′.Obviously,Δ2≥0 if and only if αα′−α−bα−b′≥0; thus, the following results hold.Theorem 8.
The modified two-parameter estimatorα^MTPEk,d,b is superior to the two-parameter estimator α^TPEk,d in the MSEM sense if and only if αα′−α−bα−b′≥0.


## 3.1. Comparison between the MTPE and OLS Using MSEM Criterion

From the representationα^MTPEk,d,b=Λ+kI−1Λ+kdIΛ−1Z′y+k1−db, the bias vector and covariance matrix of MTPE are obtained as follows:(15)Eα^MTPEk,d,b=EΛ+kI−1Λ+kdIΛ−1Z′y+k1−db=Λ+kI−1Λ+kdIΛ−1Z′Zα+k1−db=Λ+kI−1Λ+kdIα+k1−db=Λ+kI−1Λ+kdIα+Λ+kI−1k1−db,where Ey=Zα.Recall thatk1−d=Λ+kI−Λ+kdI and let Bk,d,b=Λ+kI−1Λ+kdI. Therefore,(16)Eα^MTPEk,d,b=Bk,d,bα+I−Bk,d,bb,biasα^MTPEk,d,b=Bk,d,bα+I−Bk,d,bb−α,=Bk,d,bα−Iα−b,(17)Covα^MTPEk,d,b=σ2Bk,d,bΛ−1Bk,d,b′.Hence,(18)MSEMα^MTPEk,d,b=σ2Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iα−bα−b′Bk,d,b−I′.From the representation,α^=Λ−1Z′Y, the MSEM of OLS is given as(19)MSEMα^OLS=σ2Λ−1.Comparing (18) and (19),(20)MSEMα^OLS−MSEMα^MTPEk,d,b=σ2Λ−1−Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iα−bα−b′Bk,d,b−I′.Letk > 0 and 0 < d < 1. Thus, the following theorem holds.Theorem 3.
Consider two biased competing homogenous linear estimatorsα^OLS and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to estimator α^ using the MSEM criterion, that is, MSEMα^OLS−MSEMα^MTPEk,d,b>0 if and only if(21)α−b′Bk,d,b−I′σ2Λ−1−Bk,d,bΛ−1Bk,d,b′−1Bk,d,b−Iα−b<1.Proof.
Using (17) and (19), the following was obtained:(22)Covα^−Covα^MTPEk,d,b=σ2Λ−1−Bk,d,bΛ−1Bk,d,b′=σ2diag1λi−λi+kd2λiλi+k2i=1p.Λ−1−Bk,d,bΛ−1Bk,d,b′ will be positive definite (pd) if and only if λi+k2−λi+kd2>0 or λi+k−λi+kd>0. It was observed that λi+k−λi+kd=k1−d>0 for 0<d<1 and k > 0. Therefore, Λ−1−Bk,d,bΛ−1Bk,d,b′ is pd. By Lemma 2, the proof is completed.


## 3.2. Comparison between the MTPE and RRE Using MSEM Criterion

From the representation,α^RREk=Λ+kI−1Z′y, the bias vector and covariance matrix of RRE is given as follows:(23)biasα^RREk=−kΛ+kI−1α,Covα^RREk=σ2Λ+kI−1ΛΛ+kI−1.Hence,(24)MSEMα^RREk=σ2BkΛBk′+k2Bkαα′Bk′,where Bk=Λ+kI−1. The difference between α^RREk and α^MTPEk,d,b in the MSEM sense is as follows:(25)MSEMα^RREk−MSEMα^MTPEk,d,b=σ2BkΛBk′−Bk,d,bΛ−1Bk,d,b′+k2Bkαα′Bk−Bk,d,b−Iα−bα−b′Bk,d,b−I′.Letk > 0 and 0 < d < 1. Thus, the following theorem holds.Theorem 4.
Consider two biased competing homogenous linear estimatorsα^RREk and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to estimator α^RREk using the MSEM criterion, that is, MSEMα^RREk−MSEMα^MTPEk,d,b>0 if and only if(26)α−b′Bk,d,b−I′σ2BkΛBk′−Bk,d,bΛ−1Bk,d,b′+k2Bkαα′Bk−1Bk,d,b−Iα−b<1.


## 3.3. Comparison between the MTPE and LE Using MSEM Criterion

From the representation,α^LEd=Λ+I−1Λ+dIΛ−1Z′y, the bias vector and covariance matrix of RRE are provided as follows:(27)biasα^LEd=Bd−Iα,(28)Covα^LEd=σ2BdΛ−1Bd′.Hence,(29)MSEMα^LEd=σ2BdΛ−1Bd′+Bd−Iαα′Bd−I′,where Bd=Λ+I−1Λ+dI. Considering the difference between (18) and (29),(30)MSEMα^LEd−MSEMα^MTPEk,d,b=σ2D+b1b1′−b2b2′,where D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′, b1=Bd−Iα, and b2=Bk,d,b−Iα−b.Theorem 5.
Consider two biased competing homogenous linear estimatorsα^LEd and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to estimator α^LEd using the MSEM criterion, that is, MSEMα^LEd−MSEMα^MTPEk,d,b>0 if and only if(31)α−b′Bk,d,b−I′σ2BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′+Bd−Iαα′Bd−I′−1Bk,d,b−Iα−b<1.Proof.
Using (17) and (28), the following was obtained:
By computation,(32)σ2D=σ2BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′=σ2diagλi+d2λiλi+12−λi+kd2λiλi+k2.
B
d
Λ
−
1
B
d
′
−
B
k
,
d
,
b
Λ
−
1
B
k
,
d
,
b
′ will be positive definite (pd) if and only if, λi+d2λi+k2−λi+12λi+kd2>0. For 0<d<1 and k > 1, it was observed that λi+d2λi+k2−λi+12λi+kd2>0. Therefore, BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′ is pd. By Lemma 2, the proof is completed.


## 3.4. Comparison between the MTPE and MRRE Using MSEM Criterion

From the representation,α^MRREk,b=Λ+kI−1Z′y+kb, the bias vector and covariance matrix of MRRE are provided as follows:(33)biasα^MRREk,b=Bk−Iα−b,(34)Covα^MRREk,b=σ2BkΛ−1Bk′.Hence,(35)MSEMα^MRREk,b=σ2BkΛ−1Bk′+Bk−Iα−bα−b′Bk−I′,where Bk=Λ+kI−1. Considering the difference between (18) and (35),(36)MSEMα^MRREk−MSEMα^MTPEk,d,b=σ2BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′+Bk−Iα−bα−b′Bk−I′−Bk,d,b−Iα−bα−b′Bk,d,b−I′.Theorem 6.
Consider two biased competing homogenous linear estimatorsα^MRREk,b and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to the estimator α^MRREk,b using the MSEM criterion, that is, MSEMα^MRREk,b−MSEMα^MTPEk,d,b>0 if and only if(37)Bk,d,b−I′α−b′σ2BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′+Bk−Iα−bα−b′Bk−I′−1Bk,d,b−Iα−b<1.Proof.
Using (17) and (34), the following was obtained:(38)σ2BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′=σ2diagλiλi+k2−λi+kd2λiλi+k2.
Evidently, for0<d<1 and k > 0, BkΛ−1Bk′−Bk,d,bΛ−1Bk,d,b′ will be positive definite (pd). By Lemma 2, the proof is completed.


## 3.5. Comparison between the MTPE and MLE Using MSEM Criterion

From the representation,α^MLEd,b=Λ+I−1Λ+dIΛ−1Z′y+1−db, the bias vector and covariance matrix of MLE are provided as follows:(39)biasα^MLEd=Bd−Iα−b,(40)Covα^MLEd=σ2BdΛ−1Bd′.Hence,(41)MSEMα^MLEd=σ2BdΛ−1Bd′+Bd−Iα−bα−b′Bd−I′,where Bd=Λ+I−1Λ+dI. The mean square error difference between (18) and (41) is given as(42)Δ1=MSEMα^MLEd−MSEMα^MTPEk,d,b=σ2D+b1b1′−b2b2′,where D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′, b1=Bd−Iα−b, b2=Bk,d,b−Iα−b.Theorem 7.
Consider two biased competing homogenous linear estimatorsα^MLEd,b and α^MTPEk,d,b. If k > 0 and 0<d<1, the estimator α^MTPEk,d,b is superior to the estimator α^MLEd,b using the MSEM criterion, that is, MSEMα^MLEd,b−MSEMα^MTPEk,d,b>0 if and only if b2′σ2D+b1b1′−1b2<1, where D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′, b1=Bd−Iα−b, and b2=Bk,d,b−Iα−b.Proof.
Using (17) and (40), the following was obtained:
By computation,(43)D=BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′=Qdiagτ1,…,τpQ′.
By computation,(44)σ2D=σ2BdΛ−1Bd′−Bk,d,bΛ−1Bk,d,b′=σ2diagλi+d2λiλi+12−λi+kd2λiλi+k2.
σ
2
B
d
Λ
−
1
B
d
′
−
B
k
,
d
,
b
Λ
−
1
B
k
,
d
,
b
′ will be positive definite if and only if λi+d2λi+k2−λi+kd2λi+12>0.


## 3.6. Comparison between the MTPE and TPE Using MSEM Criterion

From the representationα^TPEk,d=Λ+kI−1Λ+kdIΛ−1Z′y, the bias vector and covariance matrix of TPE are provided as follows:(45)biasα^TPEk,d=Bk,d,bα−Iα,(46)Covα^TPEk,d=σ2Bk,d,bΛ−1Bk,d,b′.Hence,(47)MSEMα^TPEk,d=σ2Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iαα′Bk,d,b−I′.Considering the matrix difference between (18) and (47)(48)Δ2=MSEMα^TPEk,d−MSEMα^MTPEk,d,b=Bk,d,b−Iαα′−α−bα−b′Bk,d,b−I′.Obviously,Δ2≥0 if and only if αα′−α−bα−b′≥0; thus, the following results hold.Theorem 8.
The modified two-parameter estimatorα^MTPEk,d,b is superior to the two-parameter estimator α^TPEk,d in the MSEM sense if and only if αα′−α−bα−b′≥0.


## 4. Selection of Bias Parameters

Selecting an appropriate parameter is crucial in this study. The use of the Ridge estimator largely depends on the ridge parameter,k. Several methods for estimating this ridge parameter have been proposed. This includes Hoerl and Kennard [1], Kibria [11], Muniz and Kibria [12], Aslam [13], Dorugade [14], Kibria and Banik [15], Lukman and Ayinde [16], Lukman et al. [17], and others. For the purpose of practical application of this new estimator, the optimum values of k and d are obtained. In order to obtain an optimum value of k, we assume the value of d is fixed.Recall from equation (18),(49)Δ=MSEMα^MTPEk,d,b=σ2Bk,d,bΛ−1Bk,d,b′+Bk,d,b−Iα−bα−b′Bk,d,b−I′=σ2∑i=1pλi+kd2λiλi+k2+k2d−12∑i=1pαi−b2λi+k2.Differentiating equation (49) with respect to k gives the following result:(50)∂Δ∂k=−2σ2∑i=1pλi1−dλi+kdλiλi+k3+2kd−12∑i=1pλiαi−b2λi+k3.Let∂Δ/∂k=0, the value of k is as follows:(51)k=σ2λiλiαi−b2−dλiαi−b2+σ2,σ2 and αi are replaced by their unbiased estimators σ^2 and α^i. The harmonic mean version is defined as(52)k^HMP=p∑i=1p1/k^,where k^=σ^2λi/λiα^i−b2−d^λiα^i−b2+σ^2.Recall thatα^MTPEk,0,0=α^RREk, considering this special case implies that k^ in equation (51) will become(53)k^=σ^2α^i2,which is the estimated value of k introduced by Hoerl and Kennard [1]. Hoerl et al. [18] defined the harmonic version of the ridge parameter, k, as follows:(54)k^HKB=pσ^2∑ipα^i2.The optimum value ofd is obtained by differentiating equation (49) with respect to d with fixed k. The result is as follows:(55)∂Δ∂d=2kσ2∑i=1pλi+kdλiλi+k2+2k2d−1∑i=1pαi−b2λi+k2.Let∂Δ/∂d=0, the value of d is as follows:(56)dMTPE=∑i=1pkλiα−b2−σ2λi∑i=1pσ2k+kλiα−b2.σ2 and αi are replaced by their unbiased estimators σ^2 and α^i. Recall that α^MTPE1,d,0=α^LEk, considering this special case implies that d in equation (54) will become(57)dliu=∑i=1pαi2−σ^2λi∑i=1pσ2+λiαi2.Equation (57) is the same as the optimum value of d proposed by Liu [6], which is defined as follows:(58)dopt=∑i=1pαi2−σ^2/λi+12∑i=1pσ2+λiαi2/λiλi+12.Theorem 9.
If(59)d^<minλiαi−b2λiαi−b2+σ2,for all i, then k^ are always positive.Proof.
The values ofk in (51) are always positive if σ2λi/λiαi−b2−dλiαi−b2+σ2>0. Since σ2λi>0,λiαi−b2−dλiαi−b2+σ2 must be positive for all i, it is observed that d<λiαi−b2/λiαi−b2+σ2 for all i. This inequality depends on the unknown parameters σ2 and αi which is replaced by their unbiased estimators σ^2 and α^i.
The selection of the estimator of the parametersd and k in α^MTPEk,d,b can be obtained iteratively as follows:Step 1: calculated^ from (59).Step 2: estimatek^HKB by using d^ in step 1.Step 3: estimated^MTPE from (56) by using the estimator k^HKB in step 2.Step 4: ifd^MTPE is negative use d^MTPE=d^, d^MTPE can take negative value. However, d^ takes value between 0 and 1.


## 5. Numerical Example and Monte-Carlo Simulation

Hussain dataset which was originally adopted by Eledum and Zahri [19] is used in this study to illustrate the performance of the new estimator. The dataset was also adopted in the study of Lukman et al. [20]. This is provided in Table 1. The regression model is defined as follows:(60)yi=β0+β1X1+β2X2+β3X3+εi,where yi represents the product value in the manufacturing sector, X1 the values of the imported intermediate commodities, X2 imported capital commodities, X3 represents the value of imported raw materials. The variance inflation factors are VIF1=128.29, VIF2=103.43, and VIF3=70.87. λ4=105.419 and the condition number of X′X is approximately 5660049. The variance inflation factor and the condition number both indicate the presence of severe multicollinearity.Table 1
y
X
1
X
2
X
3
115.20
38.10
30.40
7.00
134.30
39.20
32.40
12.50
151.00
36.30
31.40
2.30
169.00
31.10
28.40
3.60
170.80
40.00
31.40
7.00
187.50
55.00
37.00
6.00
205.20
55.00
50.00
4.00
235.70
47.00
42.00
8.00
257.70
47.00
28.10
8.70
276.70
50.00
44.70
4.50
327.00
69.00
50.00
8.50
353.80
85.00
61.40
39.20
419.50
88.00
76.10
17.70
489.00
91.00
88.70
32.90
594.90
285.00
203.30
121.00
807.60
448.00
615.90
133.90
1014.00
324.00
562.10
82.50
1208.00
281.00
716.00
99.30
1380.00
349.00
771.30
103.90
1518.00
508.40
807.40
87.70
1763.00
533.20
1222.00
217.10
1914.00
592.80
1188.00
184.90
2338.00
726.40
1478.00
227.90
2275.00
706.30
1434.00
221.40
2562.00
796.70
1630.00
250.50
2750.00
856.00
1759.00
269.50
3000.00
934.90
1930.00
294.90
2859.00
890.30
1833.00
280.60
3794.00
1185.00
2472.00
375.20
4848.00
1696.00
3581.00
539.50
4048.00
1458.00
3065.00
463.00The prior information ofb = 0.95 β^ as used in the study of Li and Yang [7] is adopted. The estimated mean square values of the estimators OLSE, RRE, LE, MRRE, MLE, TPE, and MTPE are provided in Table 2. The values of k and d were computed using the estimators of k and d proposed in this study. k and d in equations (52) and (56) are obtained to be 1036.427 and 0.0043, respectively. From both tables, OLSE has the least performance among all the estimators. It was observed from Table 2 that the modified estimators (MLE, MRRE, and MTPE) outperform their counterparts. However, the proposed estimator MTPE outperforms other estimators.Table 2
Estimated regression coefficients and mean square error of estimators.
Estimates
Estimators
OLSE
RRE
LE
MRRE
MLE
TPE
MTPE
β
^
0
208.87
207.12
191.96
198.54
208.87
207.13
195.81
β
^
1
−1.314
−1.314
−1.314
−1.314
−1.314
−1.314
−1.314
β
^
2
1.515
1.515
1.515
1.514
1.515
1.515
1.514
β
^
3
−2.017
−2.017
−2.006
−2.006
−2.017
−2.017
−2.004
MSE
1850.48
1822.65
1849.39
109.08
1564.06
1822.76
108.37Also, we conducted a Monte-Carlo simulation study to examine the performances of the estimators further. The simulation procedure used by Lukman and Ayinde [16] was also used to generate the explanatory variables in this study. This is given as(61)Xij=1−γ21/2zij+γzip,i=1,2,…,n,j=1,2,…,p,where zij is independent standard normal distribution with mean zero and unit variance, γ2 is the correlation between any two explanatory variables, and p is the number of explanatory variables. The values of γ were taken as 0.85, 0.9, and 0.99, respectively. In this study, the number of explanatory variable (p) was taken to be four.The dependent variable is generated as follows:(62)yi=β1X1+β2X2+β3X3+β4X4+εi,where εi∼0,σ2. The parameter values were chosen such that β′β = 1 which is a common restriction in simulation studies of this type [16]. The values of β are taken to be β1 = 0.8, β2 = 0.1, and β3 = 0.6. Sample sizes 50 and 100 were used. Three different values of σ (0.01, 0.1, and 1) were also used. The experiment is replicated 5000 times. The estimated MSE is calculated as(63)MSEβ^=15000∑j=15000β^ij−βi′β^ij−βi,where β^ij denotes the estimate of the ith parameter in the jth replication and βi is the true parameter values. The estimated MSEs of the estimators for different values of n, p, σ, and γ are shown in Tables 3–6. The results from the simulation study show that the estimated MSE increases as the level of error variance increases. We observed that as the degree of multicollinearity (ρ) increases, the estimated MSEs also increase. Also, RRE, MRRE, LE, MLE, TPE, and MTPE have smaller MSE than the OLS estimator. The proposed estimator MTPE outperforms other estimators depending on the choice of prior information. The results of the simulation study support the real-life analysis in this paper.Table 3
Estimated MSE values of the OLSE, RRE, MRRE, LE, and MLE whenn = 50.
Estimators
ρ = 0.85
k = 0.01
k = 0.05
k = 0.1
σ = 0.01
σ = 0.1
σ = 1
σ = 0.01
σ = 0.1
σ = 1
σ = 0.01
σ = 0.1
σ = 1
OLS
3.46E − 05
0.003874
0.333639
3.46E − 05
0.003874
0.333639
3.46E − 05
0.003874
0.333639
RRE
2.52E − 05
0.003185
0.311938
3.14E − 05
0.003145
0.319639
3.08E − 05
0.002959
0.297564
MRRE
1.57E − 05
1.98E − 03
1.87E − 01
1.95E − 05
1.95E − 03
1.99E − 01
1.91E − 05
1.84E − 03
2.37E − 01
ρ = 0.9
OLS
7.76E − 05
0.007438
0.719922
7.76E − 05
0.007438
0.719922
7.76E − 05
0.007438
0.719922
RRE
6.79E − 05
0.006505
0.629549
6.26E − 05
0.006321
0.629699
5.89E − 05
0.00608
0.579048
MRRE
4.22E − 05
4.04E − 03
3.78E − 01
3.89E − 05
3.93E − 03
3.91E − 01
3.66E − 05
3.78E − 03
4.61E − 01
ρ = 0.99
OLS
0.004043
0.408836
40.89173
0.004043
0.408836
40.89173
0.004043
0.408836
40.89173
RRE
0.002534
0.26822
24.79705
0.001099
0.109383
10.50059
0.00064
0.062891
6.359499
MRRE
0.001571
0.166296
15.37417
0.000681
0.067817
6.510364
0.000397
0.038992
3.942889
Estimators
ρ = 0.85
d = 0.01
d = 0.05
d = 0.1
0.01
0.1
1
0.01
0.1
1
0.01
0.1
1
OLS
3.46E − 05
0.003874
0.333639
3.46E − 05
0.003874
0.333639
3.46E − 05
0.003874
0.333639
LE
3.15E − 05
2.02E − 03
1.92E − 01
2.97E − 05
2.00E − 03
2.11E − 01
3.03E − 05
2.18E − 03
1.96E − 01
MLE
1.96E − 05
1.25E − 03
1.20E − 01
1.85E − 05
1.25E − 03
1.31E − 01
1.88E − 05
1.35E − 03
1.22E − 01
ρ = 0.9
OLS
7.76E − 05
0.007438
0.719922
7.76E − 05
0.007438
0.719922
7.76E − 05
0.007438
0.719922
LE
4.75E − 05
3.31E − 03
3.30E − 01
4.66E − 05
3.43E − 03
3.23E − 01
4.25E − 05
3.39E − 03
3.43E − 01
MLE
2.95E − 05
2.05E − 03
1.98E − 01
2.89E − 05
2.13E − 03
2.01E − 01
2.64E − 05
2.11E − 03
2.73E − 01
ρ = 0.99
OLS
0.004043
0.408836
40.89173
0.004043
0.408836
40.89173
0.004043
0.408836
40.89173
LE
4.45E − 05
3.00E − 03
2.79E − 01
6.72E − 05
5.84E − 03
5.25E − 01
1.00E − 04
8.96E − 03
9.42E − 01
MLE
2.76E − 05
1.86E − 03
1.68E − 01
4.17E − 05
3.63E − 03
3.26E − 01
6.22E − 05
5.57E − 03
7.50E − 01Table 4
Estimated MSE values of the OLSE, TPE, and MTPE whenn = 50.
Rho
0.85
0.9
0.99
d
k
Sigma
OLSE
TPE
MTPE
OLSE
TPE
MTPE
OLSE
TPE
MTPE
0.01
0.01
0.01
3.46E − 05
2E − 05
9.13E − 06
7.76E − 05
4.14E − 05
1.9E − 05
0.004043
0.00212
0.000969
0.1
0.003874
0.001977
0.000181
0.007438
0.003827
0.00035
0.408836
0.22438
0.020514
1
0.333639
0.188943
0.001727
0.719922
0.382846
0.0035
40.89173
20.74408
0.189653
0.05
0.01
3.46E − 05
1.99E − 05
9.09E − 06
7.76E − 05
3.78E − 05
1.7E − 05
0.004043
0.000919
0.00042
0.1
0.003874
0.00189
0.000173
0.007438
0.003928
0.00036
0.408836
0.091505
0.008366
1
0.333639
0.190314
0.00174
0.719922
0.384937
0.00352
40.89173
8.784307
0.080311
0.1
0.01
3.46E − 05
1.86E − 05
8.52E − 06
7.76E − 05
3.73E − 05
1.7E − 05
0.004043
0.000535
0.000245
0.1
0.003874
0.002014
0.000184
0.007438
0.003826
0.00035
0.408836
0.052611
0.00481
1
0.333639
0.196492
0.001796
0.719922
0.367537
0.00336
40.89173
5.320063
0.048639
0.05
0.01
0.01
3.46E − 05
1.89E − 05
8.64E − 06
7.76E − 05
4.08E − 05
1.9E − 05
0.004043
0.002114
0.000966
0.1
0.003874
0.001943
0.000178
0.007438
0.003912
0.00036
0.408836
0.224209
0.020498
1
0.333639
0.205892
0.001882
0.719922
0.368826
0.00337
40.89173
20.64197
0.18872
0.05
0.01
3.46E − 05
1.98E − 05
9.03E − 06
7.76E − 05
3.78E − 05
1.7E − 05
0.004043
0.000961
0.000439
0.1
0.003874
0.001967
0.00018
0.007438
0.003769
0.00034
0.408836
0.09861
0.009015
1
0.333639
0.186682
0.001707
0.719922
0.397967
0.00364
40.89173
9.495418
0.086812
0.1
0.01
3.46E − 05
1.97E − 05
9.02E − 06
7.76E − 05
3.96E − 05
1.8E − 05
0.004043
0.000585
0.000267
0.1
0.003874
0.001951
0.000178
0.007438
0.003655
0.00033
0.408836
0.060326
0.005515
1
0.333639
0.205296
0.001877
0.719922
0.353004
0.00323
40.89173
5.839301
0.053386
0.1
0.01
0.01
3.46E − 05
1.84E − 05
8.4E − 06
7.76E − 05
3.69E − 05
1.7E − 05
0.004043
0.002045
0.000935
0.1
0.003874
0.001917
0.000175
0.007438
0.003819
0.00035
0.408836
0.209985
0.019198
1
0.333639
0.191645
0.001752
0.719922
0.385686
0.00353
40.89173
22.09732
0.202026
0.05
0.01
3.46E − 05
1.95E − 05
8.91E − 06
7.76E − 05
3.84E − 05
1.8E − 05
0.004043
0.001
0.000457
0.1
0.003874
0.001859
0.00017
0.007438
0.003834
0.00035
0.408836
0.102129
0.009337
1
0.333639
0.192989
0.001764
0.719922
0.365375
0.00334
40.89173
10.09752
0.092317
0.1
0.01
3.46E − 05
2.01E − 05
9.19E − 06
7.76E − 05
3.56E − 05
1.6E − 05
0.004043
0.000656
0.0003
0.1
0.003874
0.002062
0.000189
0.007438
0.003699
0.00034
0.408836
0.058421
0.005341
1
0.333639
0.18755
0.001715
0.719922
0.366358
0.00335
40.89173
6.172316
0.056431Table 5
Estimated MSE values of the OLSE, RRE, MRRE, LE, and MLE whenn = 100.
ρ = 0.85
k = 0.01
k = 0.05
k = 0.1
σ = 0.01
σ = 0.1
σ = 1
σ = 0.01
σ = 0.1
σ = 1
σ = 0.01
σ = 0.1
σ = 1
OLS
0.00578
0.54722
0.57255
0.00578
0.547223
0.57255
0.005784
0.547223
0.572553
RRE
1.77E − 05
0.00176
0.16786
1.77E − 05
0.001679
0.16908
1.66E − 05
0.001789
0.174567
MRRE
1.63E − 05
0.0017
0.07039
7.42E − 06
0.000254
0.07131
8.04E − 06
0.000848
0.066346
ρ = 0.9
OLS
9.19887
8.5034
8.50041
9.19887
8.5034
8.50041
9.198873
8.5034
8.500411
RRE
3.68E − 05
0.0034
0.34013
3.36E − 05
0.00349
0.34198
3.31E − 05
0.003399
0.326526
MRRE
2.76E − 05
2.55E − 03
2.55E − 01
2.52E − 05
2.62E − 03
2.56E − 01
2.48E − 05
2.55E − 03
2.45E − 01
ρ = 0.99
OLS
1.80265
171.595
169.891
1.80265
171.595
169.891
1.802645
171.595
169.8914
RRE
0.00188
0.19934
18.4294
0.00082
0.081294
7.80411
0.000476
0.046741
4.726424
MRRE
0.00117
0.12359
11.4262
0.00051
0.050402
4.83855
0.000295
0.028979
2.930383
Sigma
ρ = 0.85
d = 0.01
d = 0.05
d = 0.1
0.01
0.1
1
0.01
0.1
1
0.01
0.1
1
OLSE
0.00578
0.54722
0.57255
0.00578
0.547223
0.57255
0.005784
0.547223
0.572553
LE
2.34E − 05
1.50E − 03
1.43E − 01
2.21E − 05
1.49E − 03
1.57E − 01
2.25E − 05
1.62E − 03
1.46E − 01
MLE
1.45E − 05
9.32E − 04
8.88E − 02
1.37E − 05
9.25E − 04
9.75E − 02
1.40E − 05
1.01E − 03
9.07E − 02
ρ = 0.9
OLSE
9.19887
8.5034
8.50041
9.19887
8.5034
8.50041
9.198873
8.5034
8.500411
LE
3.53E − 05
2.46E − 03
2.45E − 01
3.46E − 05
2.55E − 03
2.40E − 01
3.16E − 05
2.52E − 03
2.55E − 01
MLE
2.19E − 05
1.53E − 03
1.47E − 01
2.15E − 05
1.58E − 03
1.49E − 01
1.96E − 05
1.57E − 03
2.03E − 01
ρ = 0.99
OLSE
1.80265
171.595
169.891
1.80265
171.595
169.891
1.802645
171.595
169.8914
LE
3.31E − 05
2.23E − 03
2.08E − 01
4.99E − 05
4.34E − 03
3.90E − 01
7.44E − 05
6.66E − 03
7.00E − 01
MLE
2.05E − 05
1.39E − 03
1.25E − 01
3.10E − 05
2.70E − 03
2.42E − 01
4.62E − 05
4.14E − 03
5.57E − 01Table 6
Estimated MSE values of the OLSE, TPE, and MTPE whenn = 100.
d
k
σ
ρ = 0.85
ρ = 0.9
ρ = 0.99
OLSE
TPE
MTPE
OLSE
TPE
MTPE
OLSE
TPE
MTPE
0.01
0.01
0.01
5.78E − 03
1.77E − 05
8.11E − 06
9.20E + 00
3.68E − 05
1.68E − 05
1.80E + 00
1.88E − 03
8.61E − 04
0.1
5.47E − 01
1.76E − 03
1.61E − 04
8.50E + 00
3.40E − 03
3.11E − 04
1.72E + 02
1.99E − 01
1.82E − 02
1
5.73E − 01
1.68E − 01
1.54E − 03
8.50E + 00
3.40E − 01
3.11E − 03
1.70E + 02
1.84E + 01
1.68E − 01
0.05
0.01
5.78E − 03
1.77E − 05
8.08E − E − 06
9.20E + 00
3.36E − 05
1.54E − 05
1.80E + 00
8.16E − 04
3.73E − 04
0.1
5.47E − 01
1.68E − 03
1.53E − 04
8.50E + 00
3.49E − 03
3.19E − 04
1.72E + 02
8.13E − 02
7.43E − 03
1
5.73E − 01
1.69E − 01
1.55E − 03
8.50E + 00
3.42E − 01
3.13E − 03
1.70E + 02
7.80E + 00
7.13E − 02
0.1
0.01
5.78E − 03
1.66E − 05
7.57E − 06
9.20E + 00
3.31E − 05
1.51E − 05
1.80E + 00
4.76E − 04
2.17E − 04
0.1
5.47E − 01
1.79E − 03
1.64E − 04
8.50E + 00
3.40E − 03
3.11E − 04
1.72E + 02
4.67E − 02
4.27E − 03
1
5.73E − 01
1.75E − 01
1.60E − 03
8.50E + 00
3.27E − 01
2.99E − 03
1.70E + 02
4.73E + 00
4.32E − 02
0.05
0.01
0.01
5.78E − 03
1.68E − 05
7.68E − 06
9.20E + 00
3.63E − 05
1.66E − 05
1.80E + 00
1.88E − 03
8.58E − 04
0.1
5.47E − 01
1.73E − 03
1.58E − 04
8.50E + 00
3.48E − 03
3.18E − 04
1.72E + 02
1.99E − 01
1.82E − 02
1
5.73E − 01
1.83E − 01
1.67E − 03
8.50E + 00
3.28E − 01
3.00E − 03
1.70E + 02
1.83E + 01
1.68E − 01
0.05
0.01
5.78E − 03
1.76E − 05
8.03E − 06
9.20E + 00
3.36E − 05
1.54E − 05
1.80E + 00
8.54E − 04
3.90E − 04
0.1
5.47E − 01
1.75E − 03
1.60E − 04
8.50E + 00
3.35E − 03
3.06E − 04
1.72E + 02
8.76E − 02
8.01E − 03
1
5.73E − 01
1.66E − 01
1.52E − 03
8.50E + 00
3.54E − 01
3.23E − 03
1.70E + 02
8.44E + 00
7.71E − 02
0.1
0.01
5.78E − 03
1.75E − 05
8.01E − 06
9.20E + 00
3.52E − 05
1.61E − 05
1.80E + 00
5.20E − 04
2.38E − 04
0.1
5.47E − 01
1.73E − 03
1.58E − 04
8.50E + 00
3.25E − 03
2.97E − 04
1.72E + 02
5.36E − 02
4.90E − 03
1
5.73E − 01
1.82E − 01
1.67E − 03
8.50E + 00
3.14E − 01
2.87E − 03
1.70E + 02
5.19E + 00
4.74E − 02
0.1
0.01
0.01
5.78E − 03
1.63E − 05
7.46E − 06
9.20E + 00
3.28E − 05
1.50E − 05
1.80E + 00
1.82E − 03
8.30E − 04
0.1
5.47E − 01
1.70E − 03
1.56E − 04
8.50E + 00
3.39E − 03
3.10E − 04
1.72E + 02
1.87E − 01
1.71E − 02
1
5.73E − 01
1.70E − 01
1.56E − 03
8.50E + 00
3.43E − 01
3.13E − 03
1.70E + 02
1.96E + 01
1.79E − 01
0.05
0.01
5.78E − 03
1.73E − 05
7.91E − 06
9.20E + 00
3.41E − 05
1.56E − 05
1.80E + 00
8.88E − 04
4.06E − 04
0.1
5.47E − 01
1.65E − 03
1.51E − 04
8.50E + 00
3.41E − 03
3.11E − 04
1.72E + 02
9.07E − 02
8.30E − 03
1
5.73E − 01
1.71E − 01
1.57E − 03
8.50E + 00
3.25E − 01
2.97E − 03
1.70E + 02
8.97E + 00
8.20E − 02
0.1
0.01
5.78E − 03
1.79E − 05
8.16E − 06
9.20E + 00
3.16E − 05
1.44E − 05
1.80E + 00
5.83E − 04
2.66E − 04
0.1
5.47E − 01
1.83E − 03
1.68E − 04
8.50E + 00
3.29E − 03
3.00E − 04
1.72E + 02
5.19E − 02
4.75E − 03
1
5.73E − 01
1.67E − 01
1.52E − 03
8.50E + 00
3.25E − 01
2.98E − 03
1.70E + 02
5.48E + 00
5.01E − 02


## 6. Conclusions

In this article, we proposed a modified two-parameter estimator to overcome the multicollinearity problem in a linear regression model. Also, we established the superiority of this new estimator over other existing estimators in terms of matrix mean squared error criterion. This new estimator is considered to include the ordinary least squares estimator (OLSE), the ridge estimator (RRE), the Liu estimator (LE), the modified ridge estimator (MRE), and the modified Liu estimator (MLE) as special cases. Finally, a numerical example and a simulation study were conducted to illustrate the theoretical results. Results show that the performance of the proposed estimator (MTPE) is superior to others.



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*Source: 6342702-2019-05-26.xml*