New Stochastic Restricted Biased Regression Estimators

Abstract

In this paper, we propose three stochastic restricted biased estimators for the linear regression model. These new estimators generalize the least squares estimator, mixed estimator, and biased estimator. We derive the necessary and sufficient conditions for the superiority of the proposed estimators over existing ones, as well as their relative superiority among each other, using the mean squared error matrix as a criterion. A simulation study is conducted to validate the theoretical findings, and two real-world examples are provided to demonstrate the practical advantages of the proposed estimators.

1. Introduction

The statistical consequences of multicollinearity on a linear model are well-known in the field of statistics. These consequences have encouraged researchers to find a solution by improving some of the remedial approaches. One of the common remedial techniques proposed by authors is the use of biased and unbiased estimators such as the ridge estimator proposed by Hoerl and Kennard [1], the Liu estimator proposed by Liu [2], the Kibria–Lukman estimator proposed by Kibria and Lukman [3], the Dawoud–Kibria estimator (DKE) proposed by Dawoud and Kibria [4], and others. In addition, unrestricted and restricted estimators are proposed in the literature, as seen in the studies of [5,6,7,8,9,10,11,12,13,14]. Moreover, Theil [15] suggested the mixed estimator (ME) by adding stochastic linear restrictions. Then, Özkale [16] proposed the ridge estimator under stochastic restrictions. Also, Hubert and Wijekoon [17] gave a version of the Liu estimator under stochastic restrictions. Later, Yang and Xu [18] introduced another version of the Liu estimator under stochastic restrictions. In addition to that, Yang and Cui [19] gave a two-parameter estimator under stochastic restrictions. Li and Yang [20] studied the efficiency of a two-parameter estimator under stochastic restrictions. Then, Özbay and Kaçıranlar [21] studied the estimation with stochastic linear restrictions using a two-parameter weighted mixed regression estimator. Also, Tyagi and Chandra [22] obtained the two-parameter principal component estimator under stochastic restrictions. Recently, Chen and Wu [23] presented and studied the mixed Kibria–Lukman regression estimator.

The least squares estimator (LSE) is known to suffer from inflated variance in the presence of multicollinearity, while the ME incorporates prior information but does not directly address multicollinearity. The DKE effectively mitigates multicollinearity but does not account for stochastic prior information. This gap highlights the need for a unified framework that simultaneously addresses multicollinearity and incorporates stochastic restrictions. To fill this gap, we propose three stochastic restricted DKEs. These estimators generalize existing methods by integrating the principles of the DKE with stochastic restrictions. The key innovation lies in the dual ability of the proposed estimators to manage multicollinearity while incorporating prior stochastic information. We derive the necessary and sufficient conditions under which each proposed estimator outperforms the LSE, ME, and DKE, using the mean squared error matrix (MSEM) criterion. This theoretical contribution establishes a formal basis for the superiority of the proposed methods. Finally, extensive simulations and two real-life datasets are employed to demonstrate the enhanced performance of the proposed estimators compared to the LSE, ME, and DKE. The remainder of this paper is organized as follows: Section 2 introduces the statistical model and the three proposed stochastic restricted DKEs. Section 3 presents a comparative analysis of the new estimators against existing methods. Section 4 details the results of a comprehensive simulation study. In Section 5, two real-world applications are discussed to illustrate the practical relevance of the proposed estimators. Finally, Section 6 provides concluding remarks and highlights key findings.

2. Model Specification and the New Estimators

The linear regression model is as follows:

$$\tilde{y}=\tilde{X}\beta +\epsilon ,$$

(1)

where $\tilde{y}$ is an $n\times 1$ dependent vector, $\tilde{X}$ is an $n\times p$ matrix with full rank contains the explanatory variables, $\beta$ is an $p\times 1$ regression parameter vector, and $\epsilon$ is an $n\times 1$ disturbances vector with $E\left(\epsilon \right)=0$, and $Var\left(\epsilon \right)={\sigma }^2{I}_n,$ such that $I_n$ is an identity matrix. The LSE of $\beta$ in (1) is given as follows:

$$\widehat{\beta }={\tilde{S}}^{-1}{\tilde{X}}^{\prime }\tilde{y},$$

(2)

where $\tilde{S}={\tilde{X}}^{\prime }\tilde{X}$.

Providing a prior information about $\beta$ to model (1) with independent stochastic linear restrictions as follows:

$$\tilde{r}=\tilde{R}\beta +e,$$

(3)

where $\tilde{R}$ is a $j\times p$ matrix with the rank equals $j$, $e$ is a $j\times 1$ disturbances vector with $E\left(e\right)=0$ and $Var\left(e\right)={\sigma }^2\tilde{W}$, $\tilde{W}$ is known positive definite, $\tilde{r}$ is a $j\times 1$ vector with $E\left(\tilde{r}\right)=\tilde{R}\beta$, Chen and Wu [23]. The $\epsilon$ vector is stochastically independent of the vector $e$.

Combining (1), and (3), the model becomes as follows:

$${\tilde{y}}_m={\tilde{X}}_m+{\tilde{\mu }}_m,$$

(4)

where ${\tilde{y}}_m=\left(\begin{array}{c}\tilde{y}\\ \tilde{r}\end{array}\right)$, ${\tilde{X}}_m=\left(\begin{array}{c}\tilde{X}\\ \tilde{R}\end{array}\right)$, ${\tilde{\mu }}_m=\left(\begin{array}{c}\epsilon \\ e\end{array}\right)$, $E\left({\tilde{\mu }}_m\right)=0$, $Var\left({\tilde{\mu }}_m\right)={\sigma }^2\tilde{N}={\sigma }^2\left(\begin{array}{cc}I& 0\\ 0& \tilde{W}\end{array}\right)$.

The ME is obtained by minimizing ${\tilde{\Psi }}_1={\sigma }^2({\tilde{y}}_m-{\tilde{X}}_m\beta {)}^{\prime }{\sigma }^{-2}{\tilde{N}}^{-1}({\tilde{y}}_m-{\tilde{X}}_m\beta )$ with respect to $\beta$, that is given as follows:

$${\widehat{\beta }}_{ME}={\left(\tilde{S}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)}^{-1}\left({\tilde{X}}^{\prime }\tilde{y}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{r}\right),$$

(5)

also, ${\widehat{\beta }}_{ME}$ is written as follows:

$${\widehat{\beta }}_{ME}=\widehat{\beta }+{\tilde{S}}^{-1}{\tilde{R}}^{\prime }{\left(\tilde{W}+\tilde{R}{\tilde{S}}^{-1}{\tilde{R}}^{\prime }\right)}^{-1}\left(\tilde{r}-\tilde{R}\widehat{\beta }\right),$$

(6)

see Durbin [24], Theil and Goldberger [25] and Theil [15].

To treat multicollinearity, the DKE was recently proposed by Dawoud and Kibria [4] and defined as follows:

$${\widehat{\beta }}_{DKE}=\tilde{F}\widehat{\beta }=\tilde{F}{\tilde{S}}^{-1}{\tilde{X}}^{\prime }\tilde{y}={\tilde{S}}^{-1}\tilde{F}{\tilde{X}}^{\prime }\tilde{y},$$

(7)

where $\tilde{F}=(\tilde{S}+k(1+d){{\rm I}}_p{)}^{-1}(\tilde{S}-k(1+d){{\rm I}}_p)$.

• The Three Proposed Stochastic Restricted DKEs

1.

We obtain the first stochastic restricted DKE (SRDKE1) by grafting DKE into the method of mixed estimation, following Hubert and Wijekoon’s [17] procedure as follows:

$${\widehat{\beta }}_1=\tilde{F}{\widehat{\beta }}_{ME}=\tilde{F}{\left(\tilde{S}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)}^{-1}\left({\tilde{X}}^{\prime }\tilde{y}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{r}\right).$$

(8)

2.

We obtain the second stochastic restricted DKE (SRDKE2) by replacing LSE in the ME with the DKE, following Yang and Xu’s [18] procedure as follows:

$$\begin{array}{c}{\widehat{\beta }}_2={\widehat{\beta }}_{DKE}+{\tilde{S}}^{-1}{\tilde{R}}^{\prime }{\left(\tilde{W}+\tilde{R}{\tilde{S}}^{-1}{\tilde{R}}^{\prime }\right)}^{-1}(\tilde{r}-\tilde{R}{\widehat{\beta }}_{DKE})\\ ={\tilde{S}}^{-1}\tilde{F}{\tilde{X}}^{\prime }\tilde{y}+{\tilde{S}}^{-1}{\tilde{R}}^{\prime }{\left(\tilde{W}+\tilde{R}{\tilde{S}}^{-1}{\tilde{R}}^{\prime }\right)}^{-1}\left(\tilde{r}-\tilde{R}{\tilde{S}}^{-1}\tilde{F}{\tilde{X}}^{\prime }\tilde{y}\right)\\ =\left({\tilde{S}}^{-1}-{\tilde{S}}^{-1}{\tilde{R}}^{\prime }{\left(\tilde{W}+\tilde{R}{\tilde{S}}^{-1}{\tilde{R}}^{\prime }\right)}^{-1}\tilde{R}{\tilde{S}}^{-1}\right)\left(\tilde{F}{\tilde{X}}^{\prime }\tilde{y}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{r}\right)\\ ={\left(\tilde{S}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)}^{-1}\left(\tilde{F}{\tilde{X}}^{\prime }\tilde{y}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{r}\right).\end{array}$$

(9)

3.

We obtain the third stochastic restricted DKE (SRDKE3) by minimizing

$$\begin{array}{c}{\sigma }^2({\tilde{y}}_m-{\tilde{X}}_m\beta {)}^{\prime }{\sigma }^{-2}{\tilde{N}}^{-1}({\tilde{y}}_m-{\tilde{X}}_m\beta )\mathrm{subject} \mathrm{to} (\beta +\widehat{\beta }{)}^{\prime }(\beta +\widehat{\beta })=\tilde{c}, \text{as follows}:\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\sigma }^2({\tilde{y}}_m-{\tilde{X}}_m\beta {)}^{\prime }{\sigma }^{-2}{\tilde{N}}^{-1}({\tilde{y}}_m-{\tilde{X}}_m\beta )+k(1+d)[\left(\beta +\widehat{\beta }{)}^{\prime }\left(\beta +\widehat{\beta }\right)-\tilde{c}\right],\end{array}$$

(10)

where $\tilde{c}$ is a constant and $k(1+d)$ is the Lagrangian multiplier.

Now, the solution to (10) gives the SRDKE3 as follows:

$${\widehat{\beta }}_3={\left({\tilde{X}}_m^{\prime }{\tilde{N}}^{-1}{\tilde{X}}_m+k(1+d)I_p\right)}^{-1} \left({\tilde{X}}_m^{\prime }{\tilde{N}}^{-1}{\tilde{y}}_m-k(1+d)\widehat{\beta }\right), 0<d<1, k>0.$$

(11)

Equation (11) can be rewritten as follows:

$${\widehat{\beta }}_3={(\tilde{S}+k\left(1+d\right){{\rm I}}_p+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R})}^{-1}\left(\left({{\rm I}}_p-k\left(1+d\right){\tilde{S}}^{-1}\right){\tilde{X}}^{\prime }\tilde{y}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{r}\right).$$

(12)

Now, we determine the expectation and dispersion matrix of the proposed SRDKE1 as follows:

$$E({\widehat{\beta }}_1)=\tilde{F}\beta ,$$

(13)

$$Var({\widehat{\beta }}_1)={\sigma }^2\tilde{F}\tilde{A}{\tilde{F}}^{\prime },$$

(14)

where $\tilde{A}=(\tilde{S}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}{)}^{-1}$.

Now, we determine the expectation and dispersion matrix of the proposed SRDKE2 as follows:

$$E({\widehat{\beta }}_2)=\tilde{A}(\tilde{F}\tilde{S}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R})\beta ,$$

(15)

$$Var({\widehat{\beta }}_2)={\sigma }^2\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}.$$

(16)

Now, we find the expectation and dispersion matrix for the new SRDKE3 as follows:

$$E({\widehat{\beta }}_3)=\tilde{Q}(\tilde{S}-k(1+d)I_p+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R})\beta ,$$

(17)

$$Var({\widehat{\beta }}_3)={\sigma }^2\tilde{Q}\left(\tilde{S}-2k(1+d)I_p+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime },$$

(18)

where $\tilde{Q}={(\tilde{S}+k(1+d){{\rm I}}_p+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R})}^{-1}$.

3. The Comparisons of the Estimators

• The MSEM of the estimators

The MSEM of the existing estimators are as follows:

$$MSEM(\widehat{\beta })={\sigma }^2{\tilde{S}}^{-1}$$

(19)

$$MSEM({\widehat{\beta }}_{ME})={\sigma }^2\tilde{A},$$

(20)

$$MSEM\left({\widehat{\beta }}_{DKE}\right)={\sigma }^2\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime }+{\tilde{b}}_1{\tilde{b}}_1^{\prime },$$

(21)

also, we give the MSEM of the proposed estimators (SRDKE1, SRDKE2, and SRDK3) using (13)–(18) as follows:

$$MSEM({\widehat{\beta }}_1)={\sigma }^2\tilde{F}\tilde{A}{\tilde{F}}^{\prime }+{\tilde{b}}_1{\tilde{b}}_1^{\prime },$$

(22)

$$MSEM\left({\widehat{\beta }}_2\right)={\sigma }^2\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}+{\tilde{b}}_2{\tilde{b}}_2^{\prime },$$

(23)

$$MSEM\left({\widehat{\beta }}_3\right)={\sigma }^2\tilde{Q}\left(\tilde{S}-2k(1+d)I_p+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }+{\tilde{b}}_3{\tilde{b}}_3^{\prime },$$

(24)

where ${\tilde{b}}_1=Bias({\widehat{\beta }}_{DKE})=Bias({\widehat{\beta }}_1)=(\tilde{F}-I)\beta$, ${\tilde{b}}_2=Bias({\widehat{\beta }}_2)=\tilde{A}(\tilde{F}-I)\tilde{S}\beta$, ${\tilde{b}}_3=Bias({\widehat{\beta }}_3)=(\tilde{Q}\left(\tilde{S}-k\left(1+d\right)I_p+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)-I_p)\beta$.

To compare the estimators via the MSEM, we obtain the differences as follows:

$${\Delta }_1=MSEM(\widehat{\beta })-MSEM({\widehat{\beta }}_1)={\sigma }^2{\tilde{D}}_1-{\tilde{b}}_1{\tilde{b}}_1^{\prime },$$

(25)

where ${\tilde{D}}_1={\tilde{S}}^{-1}-\tilde{F}\tilde{A}{\tilde{F}}^{\prime }$.

$${\Delta }_2=MSEM({\widehat{\beta }}_{ME})-MSEM({\widehat{\beta }}_1)={\sigma }^2{D}_2-{\tilde{b}}_1{\tilde{b}}_1^{\prime },$$

(26)

where ${\tilde{D}}_2=\tilde{A}-\tilde{F}\tilde{A}{\tilde{F}}^{\prime }$.

$${\Delta }_3=MSEM({\widehat{\beta }}_{DKE})-MSEM({\widehat{\beta }}_1)={\sigma }^2{D}_3,$$

(27)

where ${\tilde{D}}_3=\tilde{F}({\tilde{S}}^{-1}-\tilde{A}){\tilde{F}}^{\prime }$.

$${\Delta }_4=MSEM(\widehat{\beta })-MSEM({\widehat{\beta }}_2)={\sigma }^2{\tilde{D}}_4-{\tilde{b}}_2{\tilde{b}}_2^{\prime },$$

(28)

where ${\tilde{D}}_4=\left({\tilde{S}}^{-1}-\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}\right)$.

$${\Delta }_5=MSEM({\widehat{\beta }}_{ME})-MSEM({\widehat{\beta }}_2)={\sigma }^2{\tilde{D}}_5-{\tilde{b}}_2{\tilde{b}}_2^{\prime },$$

(29)

where ${\tilde{D}}_5=\left(\tilde{A}-\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }\right)$.

$${\Delta }_6=MSEM({\widehat{\beta }}_{DKE})-MSEM({\widehat{\beta }}_2)={\sigma }^2{\tilde{D}}_6+{\tilde{b}}_1{\tilde{b}}_1^{\prime }-{\tilde{b}}_2{\tilde{b}}_2^{\prime },$$

(30)

where ${\tilde{D}}_6=\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime }-\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}$.

$${\Delta }_7=MSEM(\widehat{\beta })-MSEM({\widehat{\beta }}_3)={\sigma }^2{\tilde{D}}_7-{\tilde{b}}_3{\tilde{b}}_3^{\prime },$$

(31)

where ${\tilde{D}}_7=\left({\tilde{S}}^{-1}-\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\right)$.

$${\Delta }_8=MSEM({\widehat{\beta }}_{ME})-MSEM({\widehat{\beta }}_3)={\sigma }^2{\tilde{D}}_8-{\tilde{b}}_3{\tilde{b}}_3^{\prime },$$

(32)

where ${\tilde{D}}_8=\left(\tilde{A}-\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\right)$.

$${\Delta }_9=MSEM({\widehat{\beta }}_{DKE})-MSEM({\widehat{\beta }}_3)={\sigma }^2{\tilde{D}}_9+{\tilde{b}}_1{\tilde{b}}_1^{\prime }-{\tilde{b}}_3{\tilde{b}}_3^{\prime },$$

(33)

where ${\tilde{D}}_9=\left(\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime }-\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\right)$.

$${\Delta }_{10}=MSEM({\widehat{\beta }}_1)-MSEM({\widehat{\beta }}_2)={\sigma }^2{\tilde{D}}_{10}+{\tilde{b}}_1{\tilde{b}}_1^{\prime }-{\tilde{b}}_2{\tilde{b}}_2^{\prime },$$

(34)

where ${\tilde{D}}_{10}=\tilde{F}\tilde{A}{\tilde{F}}^{\prime }-\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }$.

$${\Delta }_{11}=MSEM({\widehat{\beta }}_1)-MSEM({\widehat{\beta }}_3)={\sigma }^2{\tilde{D}}_{11}+{\tilde{b}}_1{\tilde{b}}_1^{\prime }-{\tilde{b}}_3{\tilde{b}}_3^{\prime },$$

(35)

where ${\tilde{D}}_{11}=\left(\tilde{F}\tilde{A}{\tilde{F}}^{\prime }-\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\right)$.

$${\Delta }_{12}=MSEM({\widehat{\beta }}_2)-MSEM({\widehat{\beta }}_3)={\sigma }^2{\tilde{D}}_{12}+{\tilde{b}}_2{\tilde{b}}_2^{\prime }-{\tilde{b}}_3{\tilde{b}}_3^{\prime },$$

(36)

where ${\tilde{D}}_{12}=\left(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }-\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\right)$.

3.1. Comparison Between the Proposed SRDKE1 and LSE

The following theorem represents the comparison between the proposed SRDKE1 and LSE.

Theorem 1.

Under model (4), when ${\lambda }_{max}(\tilde{F}\tilde{A}{\tilde{F}}^{\prime }\tilde{S})<1$, then ${\widehat{\beta }}_1$ is superior to $\widehat{\beta }$ via the MSEM, namely ${\Delta }_1\ge 0$ iff ${\tilde{b}}_1^{\prime }{\left({\sigma }^2{\tilde{D}}_1\right)}^{-1}{\tilde{b}}_1\le 1$.

Proof. 

It is clear that ${\tilde{S}}^{-1}>0$, and $\tilde{F}\tilde{A}{\tilde{F}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{F}\tilde{A}{\tilde{F}}^{\prime }\tilde{S})<1$, then we get that ${\tilde{D}}_1>0$ by applying Lemma A1 in Appendix A. So, from (25) and applying Lemma A3 in Appendix A, we have ${\Delta }_1\ge 0$ iff ${\tilde{b}}_1^{\prime }{\left({\sigma }^2{\tilde{D}}_1\right)}^{-1}{\tilde{b}}_1\le 1$. □

3.2. Comparison Between the Proposed SRDKE1 and ME

The following theorem represents the comparison between the proposed SRDKE1 and ME.

Theorem 2.

Under model (4), when ${\lambda }_{max}(\tilde{F}\tilde{A}{\tilde{F}}^{\prime }{\tilde{A}}^{-1})<1$, then ${\widehat{\beta }}_1$ is superior to ${\widehat{\beta }}_{ME}$ via the MSEM, namely ${\Delta }_2\ge 0$ iff ${\tilde{b}}_1^{\prime }{\left({\sigma }^2{\tilde{D}}_2\right)}^{-1}{\tilde{b}}_1\le 1$.

Proof. 

It is clear that $\tilde{A}>0$, and $\tilde{F}\tilde{A}{\tilde{F}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{F}\tilde{A}{\tilde{F}}^{\prime }{\tilde{A}}^{-1})<1$, then we get that ${\tilde{D}}_2>0$ by applying Lemma A1. So, from (26) and applying Lemma A3, we have ${\Delta }_2\ge 0$ iff ${\tilde{b}}_1^{\prime }{\left({\sigma }^2{\tilde{D}}_2\right)}^{-1}{\tilde{b}}_1\le 1$. □

3.3. Comparison Between the Proposed SRDKE1 and DKE

The following theorem represents the comparison between the proposed SRDKE1 and DKE.

Theorem 3.

Under model (4), when ${\lambda }_{max}(\tilde{F}\tilde{A}\tilde{S}{\tilde{F}}^{-1})<1$, then ${\widehat{\beta }}_1$ is superior to ${\widehat{\beta }}_{DKE}$ via the MSEM, namely, ${\Delta }_3\ge 0$ iff ${\tilde{b}}_1^{\prime }{\left({\sigma }^2{\tilde{D}}_3+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_1\le 1$.

Proof. 

It is clear that $\tilde{F}{\tilde{S}}^{-1}\tilde{F}\prime >0$, and $\tilde{F}\tilde{A}\tilde{F}\prime >0$. Therefore, when ${\lambda }_{max}\left(\tilde{F}\tilde{A}\tilde{S}{\tilde{F}}^{-1}\right)<1$, then we get that ${\tilde{D}}_3>0$ by applying Lemma A1. So, from (27) and applying Lemma A3, we have ${\Delta }_3\ge 0$ iff ${\tilde{b}}_1^{\prime }{\left({\sigma }^2{\tilde{D}}_3+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_1\le 1$. □

3.4. Comparison Between the Proposed SRDKE2 and LSE

The following theorem represents the comparison between the proposed SRDKE2 and LSE.

Theorem 4.

Under model (4), when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}\tilde{S})<1$, then ${\widehat{\beta }}_2$ is superior to $\widehat{\beta }$ via the MSEM, namely, ${\Delta }_4\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_4\right)}^{-1}{\tilde{b}}_2\le 1$.

Proof. 

It is clear that ${\tilde{S}}^{-1}>0$, and $\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}>0$. Therefore, when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right)\tilde{A}\tilde{S})<1$, we get that ${\tilde{D}}_4>0$ by applying Lemma A1. So, from (28) and applying Lemma A3, we have ${\Delta }_4\ge 0$ if ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_4\right)}^{-1}{\tilde{b}}_2\le 1$. □

3.5. Comparison Between the Proposed SRDKE2 and ME

The following theorem represents the comparison between the proposed SRDKE2 and ME.

Theorem 5.

Under model (4), when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right))<1$, then ${\widehat{\beta }}_2$ is superior to ${\widehat{\beta }}_{ME}$ via the MSEM, namely, ${\Delta }_5\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_5\right)}^{-1}{\tilde{b}}_2\le 1$.

Proof. 

It is clear that $\tilde{A}>0$, and $\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right))<1$, we get that ${\tilde{D}}_5>0$ by applying Lemma A1. So, from (29) and applying Lemma A3, we have ${\Delta }_5\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_5\right)}^{-1}{\tilde{b}}_2\le 1$. □

3.6. Comparison Between the Proposed SRDKE2 and DKE

The following theorem represents the comparison between the proposed SRDKE2 and DKE.

Theorem 6.

Under model (4), when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }{(\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime })}^{-1})<1$, then ${\widehat{\beta }}_2$ is superior to ${\widehat{\beta }}_{DKE}$ via the MSEM, namely, ${\Delta }_6\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_6+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_2\le 1$.

Proof. 

It is clear that $\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime }>0$, and $\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }{(\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime })}^{-1})<1$, we get that ${\tilde{D}}_6>0$ by applying Lemma A1. So, from (30) and applying Lemma A3, we have ${\Delta }_6\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_6+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_2\le 1$. □

3.7. Comparison Between the Proposed SRDKE3 and LSE

The following theorem represents the comparison between the proposed SRDKE3 and LSE.

Theorem 7.

Under model (4), when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\tilde{S})<1$, then ${\widehat{\beta }}_3$ is superior to $\widehat{\beta }$ via the MSEM, namely, ${\Delta }_7\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_7\right)}^{-1}{\tilde{b}}_3\le 1$.

Proof. 

It is clear that ${\tilde{S}}^{-1}>0$, and $\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }\tilde{S})<1$, we get that ${\tilde{D}}_7>0$ by applying Lemma A1. So, from (31) and applying Lemma A3, we have ${\Delta }_7\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_7\right)}^{-1}{\tilde{b}}_3\le 1$. □

3.8. Comparison Between the Proposed SRDKE3 and ME

The following theorem represents the comparison between the proposed SRDKE3 and ME.

Theorem 8.

Under model (4), when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{\tilde{A}}^{-1})<1$, then ${\widehat{\beta }}_3$ is superior to ${\widehat{\beta }}_{ME}$ via the MSEM, namely, ${\Delta }_8\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_8\right)}^{-1}{\tilde{b}}_3\le 1$.

Proof. 

It is clear that $\tilde{A}>0$, and $\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{\tilde{A}}^{-1})<1$, we get that ${\tilde{D}}_8>0$ by applying Lemma A1. So, from (32) and applying Lemma A3, we have ${\Delta }_8\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_8\right)}^{-1}{\tilde{b}}_3\le 1$. □

3.9. Comparison Between the Proposed SRDKE3 and DKE

The following theorem represents the comparison between the proposed SRDKE3 and DKE.

Theorem 9.

Under model (4), when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{(\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime })}^{-1})<1$, then ${\widehat{\beta }}_3$ is superior to ${\widehat{\beta }}_{DKE}$ via the MSEM, namely, ${\Delta }_9\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_9+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_3\le 1$.

Proof. 

It is clear that $\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime }>0$, and $\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{(\tilde{F}{\tilde{S}}^{-1}{\tilde{F}}^{\prime })}^{-1})<1$, we get that ${\tilde{D}}_9>0$ by applying Lemma A1. So, from (33) and applying Lemma A3, we have ${\Delta }_9\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_9+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_3\le 1$. □

3.10. Comparison Between the Proposed SRDKE2 and the Proposed SRDKE1

The following theorem represents the comparison between the proposed SRDKE2 and the proposed SRDKE1.

Theorem 10.

Under model (4), when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }{(\tilde{F}\tilde{A}{\tilde{F}}^{\prime })}^{-1})<1$, then ${\widehat{\beta }}_2$ is superior to ${\widehat{\beta }}_1$ via the MSEM, namely, ${\Delta }_{10}\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_{10}+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_2\le 1$.

Proof. 

It is clear that $\tilde{F}\tilde{A}\tilde{F}>0$,$\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }{(\tilde{F}\tilde{A}{\tilde{F}}^{\prime })}^{-1})<1$, we get that ${\tilde{D}}_{10}>0$ by applying Lemma A1. So, from (34) and applying Lemma A3, we have ${\Delta }_{10}\ge 0$ iff ${\tilde{b}}_2^{\prime }{\left({\sigma }^2{\tilde{D}}_{10}+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_2\le 1$. □

3.11. Comparison Between the Proposed SRDKE3 and the Proposed SRDKE1

The following theorem represents the comparison between the proposed SRDKE3 and the proposed SRDKE1.

Theorem 11.

Under model (4), when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{(\tilde{F}\tilde{A}{\tilde{F}}^{\prime })}^{-1})<1$, then ${\widehat{\beta }}_3$ is superior to ${\widehat{\beta }}_1$ via the MSEM, namely, ${\Delta }_{11}\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_{11}+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_3\le 1$.

Proof. 

It is clear that $\tilde{F}\tilde{A}{\tilde{F}}^{\prime }>0$, and $\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{(\tilde{F}\tilde{A}{\tilde{F}}^{\prime })}^{-1})<1$, we get that ${\tilde{D}}_{11}>0$ by applying Lemma A1. So, from (35) and applying Lemma A3, we have ${\Delta }_{11}\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_{11}+{\tilde{b}}_1{\tilde{b}}_1^{\prime }\right)}^{-1}{\tilde{b}}_3\le 1.$ □

3.12. Comparison Between the Proposed SRDKE3 and the Proposed SRDKE2

The following theorem represents the comparison between the proposed SRDKE3 and the proposed SRDKE2.

Theorem 12.

Under model (4), when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{\left[\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }\right]}^{-1})<1$, then ${\widehat{\beta }}_3$ is superior to ${\widehat{\beta }}_2$ via the MSEM, namely, ${\Delta }_{12}\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_{12}+{\tilde{b}}_2{\tilde{b}}_2^{\prime }\right)}^{-1}{\tilde{b}}_3\le 1$.

Proof. 

It is clear that $\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }>0$, and $\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }>0$. Therefore, when ${\lambda }_{max}(\tilde{Q}\left(\tilde{S}-2k(1+d)I+k^2(1+d{)}^2{\tilde{S}}^{-1}+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{Q}}^{\prime }{\left[\tilde{A}\left(\tilde{F}\tilde{S}{\tilde{F}}^{\prime }+{\tilde{R}}^{\prime }{\tilde{W}}^{-1}\tilde{R}\right){\tilde{A}}^{\prime }\right]}^{-1})<1$, we get that ${\tilde{D}}_{12}>0$ by applying Lemma A1. So, from (36) and applying Lemma A3, we have ${\Delta }_{12}\ge 0$ iff ${\tilde{b}}_3^{\prime }{\left({\sigma }^2{\tilde{D}}_{12}+{\tilde{b}}_2{\tilde{b}}_2^{\prime }\right)}^{-1}{\tilde{b}}_3\le 1$. □

4. A Simulation Study

A simulation is performed to show the proposed (SRDKE1, SRDKE2, and SRDKE3) performances in addition to the existing ones. MATLAB software is used for the computations. We use the equation below to generate the matrix of explanatory variables.

$${\tilde{x}}_{ij}=(1-{\tilde{\gamma }}^2{)}^{1/2}{\tilde{z}}_{ij}+\tilde{\gamma }{\tilde{z}}_{i,p+1},i=\mathrm{1,2},...,n,j=\mathrm{1,2},...,p$$

(37)

where ${\tilde{z}}_{ij}$ ‘s are independent pseudo-random values which follow a standardized normal distribution, and $\tilde{\gamma }$ is chosen, where ${\tilde{\gamma }}^2$ is the correlation between two columns of $\tilde{X}$, (Kibria [26] and Dawoud and Kibria [4]). Also, we take the largest eigenvalue of $\tilde{S}$ as the parameter vector following Kibria [26].

The dependent variable $\tilde{y}$ is generated as follows:

$${\tilde{y}}_i={\beta }_1{\tilde{x}}_{i1}+{\beta }_2{\tilde{x}}_{i2}+\dots +{\beta }_p{\tilde{x}}_{ip}+e_i,i=\mathrm{1,2},\dots ,n$$

(38)

where $e_i$ ‘s are $i.i.dN(0,{\sigma }^2)$ and $\beta$ ‘s values are selected such that ${\beta }^{\prime }\beta =1$, (Dawoud and Kibria [4]).

We state the factors’ values used in the simulation in

Table 1. The factors’ values.

Factor	Symbol	Values
Sample size	$n$	30, 50, 70, 100, 200, 500
Standard deviation	$\sigma$	1, 5, 9
Degree of correlation	$\tilde{\gamma }$	0.7, 0.8, 0.9, 0.95, 0.99
Number of explanatory variables	$p$	3, 7, 10
Number of replicates	MCN	1000

.

Here, we consider the stochastic restrictions for each $p$, following the style provided by Roozbeh et al. [27] and Roozbeh and Hamzah [28], as follows:

-

For $p=3$, $\tilde{R}$ is given as $\tilde{R}=(1-2-2)$.

-

For $p=7$, $\tilde{R}$ is given as $\tilde{R}=(1-2-2-2-2-2-2)$.

-

For $p=10$, $\tilde{R}$ is given as $\tilde{R}=(1-2-2-2-2-2-2-2-2-2)$.

where $\tilde{r}=\tilde{R}\beta +e$, $e~N(0,{\sigma }_{LSE}^2)$

Also, we use the following estimators of $(k,d)$ for the DKE and SRDKE1, SRDKE2, and SRDKE3.

$$\widehat{d}={\mathit{min}\left({\displaystyle \frac{{\widehat{\alpha }}_j^2}{\frac{{\widehat{\sigma }}^2}{{\lambda }_j}+{\widehat{\alpha }}_j^2}}\right)}_{j=1}^p,$$

(39)

$$\widehat{k}={\mathit{min}\left({\displaystyle \frac{{\widehat{\sigma }}^2}{\left(1+\widehat{d}\right)\left(\frac{{\widehat{\sigma }}^2}{{\lambda }_j}+2{\widehat{\alpha }}_j^2\right)}}\right)}_{j=1}^p,$$

(40)

where ${\lambda }_j$ is j-the eigenvalue of $\tilde{S}$, $\alpha =P^{\prime }\beta$, $P$ is an orthogonal matrix, ${\widehat{\sigma }}^2$ is the estimated value of ${\sigma }^2$, $\widehat{\alpha }$ is the estimator of LSE for $\alpha$; see [4,29].

To verify the performances of the LSE, ME, DKE, and the proposed estimators (SRDKE1, SRDKE2, and SRDKE3), we use the following formula:

$$MSE({\tilde{\delta }}^{\ast })={\displaystyle \frac{1}{MCN}}{\sum }_{j=1}^{MCN}({\tilde{\delta }}_{ij}^{\ast }-{\tilde{\delta }}_i{)}^{\prime }({\tilde{\delta }}_{ij}^{\ast }-{\tilde{\delta }}_i)$$

(41)

where ${\tilde{\delta }}_{ij}^{\ast }$ is estimator and ${\tilde{\delta }}_i$ is parameter.

The results of the simulation are given in

Table 2. Estimated MSE of the estimators.

$\mathit{\sigma }=1$$, \mathit{p}=3$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	0.1314	0.1159	0.1126	0.0990	0.1012	0.1009
	0.80	0.1762	0.1486	0.1456	0.1218	0.1263	0.1256
	0.90	0.3158	0.2422	0.2391	0.1799	0.1934	0.1912
	0.95	0.5976	0.4088	0.4072	0.2696	0.3021	0.2973
	0.99	2.8540	1.4625	1.5891	0.7876	0.8875	0.8776
50	0.70	0.0972	0.0884	0.0881	0.0800	0.0809	0.0808
	0.80	0.1321	0.1160	0.1161	0.1016	0.1035	0.1033
	0.90	0.2402	0.1960	0.1955	0.1579	0.1646	0.1636
	0.95	0.4582	0.3408	0.3380	0.2457	0.2646	0.2621
	0.99	2.2053	1.2726	1.3116	0.7301	0.8062	0.8001
70	0.70	0.0626	0.0579	0.0588	0.0545	0.0547	0.0547
	0.80	0.0842	0.0754	0.0778	0.0696	0.0702	0.0701
	0.90	0.1523	0.1275	0.1347	0.1122	0.1143	0.1140
	0.95	0.2903	0.2220	0.2413	0.1818	0.1891	0.1882
	0.99	1.3981	0.8094	0.9794	0.5421	0.5901	0.5853
100	0.70	0.0428	0.0402	0.0405	0.0381	0.0384	0.0383
	0.80	0.0580	0.0532	0.0535	0.0491	0.0497	0.0496
	0.90	0.1059	0.0920	0.0909	0.0787	0.0810	0.0808
	0.95	0.2033	0.1633	0.1553	0.1234	0.1318	0.1308
	0.99	0.9869	0.6092	0.5225	0.3162	0.3718	0.3669
200	0.70	0.0225	0.0218	0.0219	0.0213	0.0213	0.0213
	0.80	0.0302	0.0289	0.0291	0.0279	0.0279	0.0279
	0.90	0.0545	0.0504	0.0509	0.0471	0.0474	0.0473
	0.95	0.1040	0.0913	0.0919	0.0805	0.0819	0.0818
	0.99	0.5012	0.3587	0.3486	0.2447	0.2686	0.2656
500	0.70	0.0080	0.0079	0.0080	0.0079	0.0079	0.0079
	0.80	0.0108	0.0106	0.0107	0.0105	0.0105	0.0105
	0.90	0.0195	0.0188	0.0191	0.0184	0.0185	0.0185
	0.95	0.0373	0.0350	0.0357	0.0334	0.0336	0.0336
	0.99	0.1798	0.1474	0.1490	0.1209	0.1262	0.1255

,

Table 3. Estimated MSE of the estimators.

$\mathit{\sigma }=1$$, \mathit{p}=7$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	0.5311	0.4041	0.5114	0.3878	0.3932	0.3923
	0.80	0.7417	0.5395	0.7076	0.5125	0.5216	0.5200
	0.90	1.3864	0.9311	1.2890	0.8614	0.8842	0.8803
	0.95	2.6825	1.6714	2.4209	1.5024	1.5523	1.5435
	0.99	13.0461	7.2661	11.2220	6.2623	6.4589	6.4177
50	0.70	0.3102	0.2466	0.3005	0.2381	0.2411	0.2403
	0.80	0.4363	0.3320	0.4171	0.3159	0.3228	0.3211
	0.90	0.8217	0.5753	0.7609	0.5289	0.5524	0.5461
	0.95	1.5966	1.0247	1.4232	0.9063	0.9671	0.9505
	0.99	7.8019	4.3092	6.4864	3.5837	3.8728	3.7879
70	0.70	0.1796	0.1550	0.1755	0.1514	0.1519	0.1518
	0.80	0.2526	0.2101	0.2449	0.2035	0.2047	0.2046
	0.90	0.4762	0.3681	0.4528	0.3493	0.3536	0.3530
	0.95	0.9267	0.6598	0.8584	0.6088	0.6213	0.6195
	0.99	4.5411	2.7198	3.9735	2.3611	2.4206	2.4130
100	0.70	0.1263	0.1125	0.1230	0.1096	0.1099	0.1098
	0.80	0.1771	0.1532	0.1712	0.1480	0.1486	0.1486
	0.90	0.3330	0.2706	0.3155	0.2560	0.2585	0.2582
	0.95	0.6469	0.4876	0.5960	0.4478	0.4562	0.4550
	0.99	3.1615	1.9927	2.7228	1.7075	1.7610	1.7525
200	0.70	0.0566	0.0529	0.0561	0.0524	0.0525	0.0525
	0.80	0.0795	0.0728	0.0784	0.0719	0.0720	0.0720
	0.90	0.1495	0.1309	0.1459	0.1276	0.1283	0.1282
	0.95	0.2905	0.2391	0.2778	0.2285	0.2310	0.2307
	0.99	1.4200	0.9727	1.2618	0.8625	0.8872	0.8845
500	0.70	0.0241	0.0233	0.0240	0.0232	0.0232	0.0232
	0.80	0.0338	0.0324	0.0336	0.0321	0.0321	0.0321
	0.90	0.0637	0.0592	0.0629	0.0585	0.0585	0.0585
	0.95	0.1238	0.1104	0.1210	0.1079	0.1081	0.1081
	0.99	0.6054	0.4631	0.5617	0.4287	0.4345	0.4340

,

Table 4. Estimated MSE of the estimators.

$\mathit{\sigma }=1$$, \mathit{p}=10$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	0.9340	0.6448	0.9201	0.6328	0.6374	0.6364
	0.80	1.3165	0.8721	1.2915	0.8511	0.8607	0.8584
	0.90	2.4838	1.5345	2.4071	1.4749	1.5043	1.4972
	0.95	4.8302	2.8163	4.6139	2.6628	2.7325	2.7158
	0.99	23.6169	12.8952	22.0274	11.9006	12.2251	12.1443
50	0.70	0.4215	0.3266	0.4133	0.3200	0.3239	0.3229
	0.80	0.5924	0.4404	0.5754	0.4277	0.4362	0.4338
	0.90	1.1156	0.7700	1.0595	0.7313	0.7576	0.7502
	0.95	2.1682	1.3941	2.0052	1.2910	1.3545	1.3370
	0.99	10.6006	6.0970	9.3621	5.4186	5.7010	5.6200
70	0.70	0.2988	0.2447	0.2953	0.2416	0.2424	0.2422
	0.80	0.4213	0.3328	0.4151	0.3274	0.3289	0.3286
	0.90	0.7957	0.5865	0.7769	0.5711	0.5756	0.5748
	0.95	1.5488	1.0618	1.4939	1.0194	1.0315	1.0295
	0.99	7.5830	4.5758	7.1131	4.2543	4.3107	4.3021
100	0.70	0.1824	0.1588	0.1792	0.1561	0.1565	0.1565
	0.80	0.2575	0.2178	0.2517	0.2129	0.2138	0.2137
	0.90	0.4868	0.3876	0.4695	0.3737	0.3769	0.3764
	0.95	0.9477	0.7049	0.8975	0.6672	0.6766	0.6752
	0.99	4.6403	2.9926	4.2121	2.7172	2.7716	2.7628
200	0.70	0.0902	0.0829	0.0893	0.0821	0.0821	0.0821
	0.80	0.1271	0.1143	0.1255	0.1128	0.1129	0.1129
	0.90	0.2401	0.2056	0.2348	0.2010	0.2017	0.2016
	0.95	0.4674	0.3765	0.4507	0.3630	0.3656	0.3653
	0.99	2.2886	1.5718	2.1106	1.4487	1.4727	1.4697
500	0.70	0.0351	0.0336	0.0351	0.0335	0.0335	0.0335
	0.80	0.0496	0.0468	0.0494	0.0466	0.0466	0.0466
	0.90	0.0936	0.0855	0.0929	0.0849	0.0850	0.0850
	0.95	0.1821	0.1590	0.1797	0.1569	0.1572	0.1572
	0.99	0.8911	0.6709	0.8514	0.6402	0.6463	0.6457

,

Table 5. Estimated MSE of the estimators.

$\mathit{\sigma }=5$$, \mathit{p}=3$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	3.2844	1.8598	2.8146	1.5517	1.7454	1.6787
	0.80	4.4059	2.3443	3.6393	1.8605	2.1835	2.0735
	0.90	7.8944	3.8175	5.9767	2.7144	3.4341	3.2112
	0.95	14.9410	6.7379	10.1812	4.2506	5.5873	5.2480
	0.99	71.3502	29.9278	39.7278	15.7586	18.6232	18.2843
50	0.70	2.4306	1.4904	2.2022	1.3314	1.4307	1.3984
	0.80	3.3031	1.9075	2.9016	1.6353	1.8110	1.7538
	0.90	6.0038	3.1630	4.8887	2.4591	2.9021	2.7649
	0.95	11.4556	5.6279	8.4489	3.8869	4.8268	4.5736
	0.99	55.1320	25.0866	32.7908	14.1142	16.5322	16.2410
70	0.70	1.5647	0.9724	1.4709	0.9089	0.9433	0.9331
	0.80	2.1044	1.2128	1.9454	1.1077	1.1705	1.1512
	0.90	3.8063	1.9505	3.3682	1.6796	1.8590	1.8034
	0.95	7.2587	3.3903	6.0321	2.6890	3.1406	3.0104
	0.99	34.9523	14.6646	24.4848	9.5281	11.2005	10.9409
100	0.70	1.0695	0.7064	1.0131	0.6617	0.6981	0.6864
	0.80	1.4491	0.8833	1.3382	0.8005	0.8697	0.8470
	0.90	2.6477	1.4214	2.2732	1.1718	1.3778	1.3099
	0.95	5.0833	2.4597	3.8813	1.7499	2.2794	2.1152
	0.99	24.6721	10.4944	13.0634	5.1245	7.0884	6.7536
200	0.70	0.5616	0.4235	0.5481	0.4130	0.4176	0.4165
	0.80	0.7546	0.5290	0.7281	0.5089	0.5194	0.5166
	0.90	1.3632	0.8436	1.2734	0.7801	0.8208	0.8086
	0.95	2.5996	1.4296	2.2964	1.2317	1.3737	1.3285
	0.99	12.5310	5.7688	8.7142	3.7397	4.8598	4.5713
500	0.70	0.2005	0.1712	0.1992	0.1701	0.1706	0.1704
	0.80	0.2698	0.2176	0.2672	0.2152	0.2163	0.2161
	0.90	0.4883	0.3539	0.4780	0.3449	0.3502	0.3488
	0.95	0.9318	0.5995	0.8919	0.5666	0.5896	0.5827
	0.99	4.4945	2.2239	3.7245	1.7331	2.1118	1.9923

,

Table 6. Estimated MSE of the estimators.

$\mathit{\sigma }=5$$, \mathit{p}=7$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	13.2784	7.4696	12.7838	7.1332	7.3680	7.3049
	0.80	18.5427	10.2637	17.6899	9.7128	10.0710	9.9776
	0.90	34.6605	18.8340	32.2249	17.3772	18.1449	17.9591
	0.95	67.0617	36.0993	60.5237	32.3962	33.8721	33.5446
	0.99	326.1516	174.1541	280.5489	150.0596	155.1024	153.9788
50	0.70	7.7551	4.3207	7.5134	4.1423	4.2913	4.2356
	0.80	10.9074	5.9449	10.4268	5.6100	5.9111	5.7974
	0.90	20.5423	10.9246	19.0224	9.9532	10.7780	10.4792
	0.95	39.9158	20.9457	35.5797	18.3852	20.2122	19.5993
	0.99	195.0467	101.3026	162.1610	84.1030	91.4428	89.1713
70	0.70	4.4909	2.7389	4.3882	2.6684	2.7082	2.6972
	0.80	6.3154	3.7356	6.1234	3.6082	3.6835	3.6630
	0.90	11.9058	6.7635	11.3200	6.3905	6.6000	6.5467
	0.95	23.1677	12.8368	21.4601	11.7872	12.2780	12.1692
	0.99	113.5283	61.3395	99.3379	53.1668	54.8679	54.6164
100	0.70	3.1574	2.0180	3.0756	1.9587	1.9842	1.9777
	0.80	4.4280	2.7366	4.2804	2.6332	2.6813	2.6687
	0.90	8.3255	4.9128	7.8882	4.6219	4.7622	4.7255
	0.95	16.1718	9.2518	14.9003	8.4461	8.8051	8.7174
	0.99	79.0385	43.7732	68.0706	37.4133	38.9856	38.6913
200	0.70	1.4151	0.9792	1.4017	0.9696	0.9764	0.9746
	0.80	1.9870	1.3174	1.9607	1.2989	1.3129	1.3091
	0.90	3.7383	2.3184	3.6466	2.2566	2.3050	2.2916
	0.95	7.2628	4.2724	6.9459	4.0684	4.2170	4.1776
	0.99	35.5011	19.5927	31.5458	17.2975	18.1896	18.0420
500	0.70	0.6026	0.4653	0.5995	0.4627	0.4633	0.4632
	0.80	0.8461	0.6258	0.8402	0.6211	0.6225	0.6223
	0.90	1.5925	1.0900	1.5725	1.0752	1.0808	1.0798
	0.95	3.0950	1.9709	3.0247	1.9219	1.9436	1.9392
	0.99	15.1350	8.6904	14.0420	8.0097	8.2815	8.2323

,

Table 7. Estimated MSE of the estimators.

$\mathit{\sigma }=5$$, \mathit{p}=10$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	23.3500	12.7717	23.0016	12.5057	12.6771	12.6305
	0.80	32.9133	17.8836	32.2863	17.4167	17.7431	17.6513
	0.90	62.0960	33.4739	60.1765	32.1398	33.0125	32.7735
	0.95	120.7539	64.8056	115.3477	61.2615	63.1611	62.6716
	0.99	590.4232	315.8951	550.6839	291.5832	299.7943	297.7157
50	0.70	10.5370	6.0964	10.3324	5.9559	6.1304	6.0615
	0.80	14.8094	8.4512	14.3851	8.1755	8.5139	8.3834
	0.90	27.8893	15.6537	26.4885	14.8047	15.6878	15.3721
	0.95	54.2060	30.1277	50.1294	27.7955	29.6871	29.0738
	0.99	265.0157	146.1229	234.0516	129.7329	137.0945	134.8982
70	0.70	7.4691	4.5248	7.3830	4.4614	4.5032	4.4917
	0.80	10.5329	6.2698	10.3765	6.1578	6.2300	6.2107
	0.90	19.8931	11.5773	19.4235	11.2511	11.4371	11.3913
	0.95	38.7200	22.2169	37.3473	21.2867	21.7105	21.6190
	0.99	189.5758	107.4084	177.8281	99.7729	101.3285	101.0731
100	0.70	4.5595	2.9542	4.4803	2.8994	2.9302	2.9222
	0.80	6.4382	4.0782	6.2930	3.9807	4.0352	4.0212
	0.90	12.1695	7.4777	11.7368	7.1983	7.3458	7.3094
	0.95	23.6925	14.2600	22.4370	13.4777	13.8373	13.7561
	0.99	116.0072	68.4443	105.3017	62.1188	63.6564	63.3757
200	0.70	2.2541	1.5591	2.2321	1.5435	1.5497	1.5484
	0.80	3.1787	2.1302	3.1369	2.1014	2.1141	2.1113
	0.90	6.0033	3.8429	5.8708	3.7545	3.7973	3.7873
	0.95	11.6850	7.2452	11.2679	6.9724	7.1018	7.0721
	0.99	57.2148	34.4361	52.7639	31.6673	32.4377	32.3162
500	0.70	0.8786	0.6654	0.8764	0.6638	0.6644	0.6643
	0.80	1.2388	0.9042	1.2344	0.9010	0.9025	0.9022
	0.90	2.3391	1.6025	2.3231	1.5913	1.5970	1.5959
	0.95	4.5516	2.9513	4.4927	2.9114	2.9327	2.9284
	0.99	22.2774	13.4680	21.2860	12.8298	13.0842	13.0424

,

Table 8. Estimated MSE of the estimators.

$\mathit{\sigma }=9$$, \mathit{p}=3$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	10.6414	5.5011	9.1194	4.5639	5.2484	4.9930
	0.80	14.2752	7.0485	11.7913	5.5652	6.6830	6.2751
	0.90	25.5777	11.7622	19.3647	8.3162	10.7433	9.9516
	0.95	48.4088	21.1865	32.9872	13.3064	17.7357	16.5731
	0.99	231.1746	96.3864	128.7182	50.7303	60.0441	58.9371
50	0.70	7.8751	4.3120	7.1351	3.8288	4.1990	4.0636
	0.80	10.7021	5.6021	9.4013	4.7713	5.4070	5.1766
	0.90	19.4522	9.5448	15.8395	7.3673	8.9069	8.3865
	0.95	37.1162	17.4196	27.3743	11.9531	15.1275	14.2193
	0.99	178.6278	80.5225	106.2423	45.2426	53.1521	52.1812
70	0.70	5.0697	2.7454	4.7658	2.5586	2.6946	2.6467
	0.80	6.8183	3.4706	6.3030	3.1606	3.3983	3.3142
	0.90	12.3325	5.7442	10.9130	4.9281	5.5690	5.3496
	0.95	23.5181	10.3147	19.5439	8.1392	9.6848	9.2098
	0.99	113.2455	46.7908	79.3309	30.3429	35.8063	34.9508
100	0.70	3.4653	1.9368	3.2824	1.8019	1.9451	1.8899
	0.80	4.6950	2.4488	4.3357	2.1984	2.4602	2.3591
	0.90	8.5786	4.0611	7.3652	3.2979	4.0348	3.7582
	0.95	16.4698	7.2979	12.5753	5.1021	6.9203	6.3020
	0.99	79.9378	33.3089	42.3253	16.1586	22.5936	21.4678
200	0.70	1.8195	1.1503	1.7759	1.1190	1.1411	1.1336
	0.80	2.4450	1.4363	2.3590	1.3771	1.4234	1.4071
	0.90	4.4168	2.3225	4.1258	2.1348	2.2978	2.2377
	0.95	8.4227	4.0777	7.4403	3.4834	4.0066	3.8134
	0.99	40.6006	17.9766	28.2339	11.5751	15.3014	14.3015
500	0.70	0.6496	0.4784	0.6455	0.4748	0.4776	0.4767
	0.80	0.8742	0.5952	0.8656	0.5875	0.5939	0.5918
	0.90	1.5820	0.9473	1.5488	0.9193	0.9445	0.9356
	0.95	3.0191	1.6070	2.8896	1.5070	1.6030	1.5671
	0.99	14.5620	6.5245	12.0675	5.0191	6.3464	5.8776

,

Table 9. Estimated MSE of the estimators.

$\mathit{\sigma }=9$$, \mathit{p}=7$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	43.0219	23.5983	41.4197	22.5239	23.3066	23.0888
	0.80	60.0783	32.6095	57.3152	30.8460	32.0331	31.7143
	0.90	112.2999	60.3731	104.4088	55.6912	58.2083	57.5885
	0.95	217.2799	116.2659	196.0969	104.3295	109.1388	108.0623
	0.99	1056.7200	563.6500	909.0100	485.7100	502.0400	498.4200
50	0.70	25.1264	13.3936	24.3434	12.8194	13.3204	13.1264
	0.80	35.3401	18.6495	33.7828	17.5665	18.5666	18.1788
	0.90	66.5569	34.7779	61.6326	31.6444	34.3410	33.3509
	0.95	129.3271	67.2589	115.2781	59.0015	64.9367	62.9344
	0.99	631.9514	327.5880	525.4015	271.9808	295.7584	288.3920
70	0.70	14.5505	8.3487	14.2176	8.1287	8.2700	8.2281
	0.80	20.4620	11.5215	19.8398	11.1209	11.3819	11.3066
	0.90	38.5749	21.2670	36.6767	20.0815	20.7871	20.6007
	0.95	75.0634	40.8703	69.5307	37.5133	39.1406	38.7703
	0.99	367.8316	197.8239	321.8548	171.4551	177.0017	176.1757
100	0.70	10.2299	6.0551	9.9649	5.8706	5.9644	5.9372
	0.80	14.3467	8.3199	13.8686	7.9965	8.1676	8.1182
	0.90	26.9745	15.2608	25.5579	14.3444	14.8235	14.6908
	0.95	52.3965	29.2195	48.2769	26.6614	27.8583	27.5561
	0.99	256.0847	140.9702	220.5487	120.4924	125.6165	124.6521
200	0.70	4.5848	2.8228	4.5416	2.7942	2.8218	2.8133
	0.80	6.4379	3.8444	6.3527	3.7887	3.8439	3.8266
	0.90	12.1122	6.9488	11.8149	6.7577	6.9353	6.8809
	0.95	23.5314	13.1417	22.5046	12.4980	13.0161	12.8692
	0.99	115.0237	62.6848	102.2085	55.3150	58.2539	57.7568
500	0.70	1.9525	1.3184	1.9423	1.3106	1.3136	1.3130
	0.80	2.7414	1.7777	2.7222	1.7637	1.7700	1.7686
	0.90	5.1597	3.1508	5.0948	3.1067	3.1295	3.1242
	0.95	10.0277	5.8703	9.8000	5.7217	5.8026	5.7833
	0.99	49.0373	27.4528	45.4962	25.3021	26.2130	26.0401

and

Table 10. Estimated MSE of the estimators.

$\mathit{\sigma }=9$$, \mathit{p}=10$
		Existing Estimators			Proposed Estimators
$\mathit{n}$	$\tilde{\mathit{\gamma }}$	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
30	0.70	75.6541	40.8268	74.5251	39.9746	40.5385	40.3827
	0.80	106.6391	57.3585	104.6077	55.8624	56.9320	56.6282
	0.90	201.1909	107.8572	194.9719	103.5650	106.4074	105.6255
	0.95	391.2426	209.4222	373.7267	197.9733	204.1437	202.5501
	0.99	1913.0100	1023.0200	1784.2500	944.3400	970.9700	964.2400
50	0.70	34.1398	19.2547	33.4769	18.8010	19.3862	19.1499
	0.80	47.9824	26.8403	46.6078	25.9484	27.0735	26.6305
	0.90	90.3614	50.1312	85.8227	47.3854	50.2895	49.2382
	0.95	175.6273	97.0345	162.4191	89.4907	95.6703	93.6534
	0.99	858.6510	473.0814	758.3271	420.0134	443.9173	436.7817
70	0.70	24.1998	14.0969	23.9211	13.8971	14.0405	13.9991
	0.80	34.1265	19.6897	33.6200	19.3343	19.5797	19.5117
	0.90	64.4535	36.7918	62.9321	35.7484	36.3697	36.2132
	0.95	125.4528	71.2306	121.0051	68.2371	69.6323	69.3275
	0.99	614.2256	347.1575	576.1630	322.4593	327.5137	326.6826
100	0.70	14.7729	9.0572	14.5163	8.8877	8.9965	8.9656
	0.80	20.8597	12.6361	20.3893	12.3311	12.5205	12.4682
	0.90	39.4292	23.5407	38.0272	22.6570	23.1550	23.0266
	0.95	76.7638	45.4563	72.6960	42.9535	44.1453	43.8687
	0.99	375.8633	220.8507	341.1775	200.4031	205.4147	204.4948
200	0.70	7.3034	4.6541	7.2321	4.6060	4.6298	4.6239
	0.80	10.2990	6.4430	10.1637	6.3533	6.4006	6.3885
	0.90	19.4507	11.9093	19.0214	11.6281	11.7793	11.7407
	0.95	37.8593	22.8995	36.5080	22.0230	22.4622	22.3571
	0.99	185.3759	110.9176	170.9550	101.9760	104.4965	104.0963
500	0.70	2.8466	1.9260	2.8394	1.9211	1.9240	1.9234
	0.80	4.0139	2.6361	3.9996	2.6266	2.6327	2.6314
	0.90	7.5786	4.7715	7.5270	4.7376	4.7598	4.7547
	0.95	14.7473	9.0224	14.5562	8.8968	8.9748	8.9571
	0.99	72.1787	42.9337	68.9666	40.8899	41.7395	41.5959

.

The performance of the estimators is influenced by several key factors:

• Sample Size ($n$): As the sample size increases, the MSE for all estimators decreases, reflecting the typical behavior where larger samples yield more stable and reliable parameter estimates.
• Degree of Correlation ($\tilde{\mathit{\gamma }}$): As the degree of correlation between explanatory variables increases, the MSE for all estimators also increases. This is expected since higher multicollinearity leads to less stable and less reliable estimates.
• Number of Explanatory Variables ($p$): An increase in the number of explanatory variables results in a higher MSE for all estimators. The increase in multicollinearity and model complexity associated with larger p explains this behavior.
• Standard Deviation of the Error Term ($\sigma$): As the standard deviation of the error term increases, the MSE for all estimators rises accordingly. A larger $\sigma$ reflects higher variability in the error term, making parameter estimation more challenging and less precise.
• LSE Performance: The LSE consistently exhibits the highest MSE across all sample sizes. While its MSE decreases as the sample size increases, it remains the least efficient estimator in the presence of multicollinearity.
• Effect of High Correlation ($\tilde{\gamma }$ ≈ 0.99): The LSE shows a sharp increase in MSE as $\tilde{\mathit{\gamma }}$ approaches 0.99, highlighting its inefficiency in high multicollinearity situations, where it amplifies the variance of parameter estimates.
• Effect of Model Complexity ($p$ = 3 to $p$ = 10): The LSE demonstrates a significant increase in MSE as $p$ increases from 3 to 9, illustrating its sensitivity to model complexity and the accompanying multicollinearity.
• Sensitivity to Noise ($\sigma$): Among the estimators, the LSE exhibits the largest increase in MSE as σ increases, indicating its sensitivity to noise in the data.
• Comparative Performance of ME, DKE, and SRDKEs: While the ME and the DKE generally outperform the LSE, they still underperform compared to the stochastic restricted DKE estimators (SRDKE1, SRDKE2, and SRDKE3) in most cases. The latter estimators demonstrate superior robustness under varying conditions of multicollinearity, model complexity, and noise.
• Superior Performance of SRDKE1: The SRDKE1 consistently outperforms all other estimators across a wide range of scenarios, including varying sample sizes, degrees of correlation, numbers of explanatory variables, and error variances.
• Robustness and Efficiency: Among the proposed estimators, SRDKE1 emerges as the most robust and efficient, particularly in settings characterized by high multicollinearity, large error variability, and complex models with many explanatory variables. While the SRDKE2 and SRDKE3 estimators also demonstrate strong performance, they are slightly less effective than SRDKE1. In contrast, the LSE remains the least robust estimator, especially in the presence of multicollinearity under stochastic restrictions.

Figure 1 illustrates the comparison of the MSE for six estimators: LSE, ME, DKE, SRDKE1, SRDKE2, and SRDKE3. As observed, the MSE for all estimators increases as $\tilde{\gamma }$, σ, and $p$ increase. Conversely, the MSE decreases as the sample size ($n$) increases. Among the estimators, the LSE demonstrates the worst performance, exhibiting the highest MSE under all conditions. The ME and DKE perform better than the LSE but are still outperformed by the proposed estimators (SRDKE1, SRDKE2, and SRDKE3). Notably, SRDKE1 achieves the lowest MSE, highlighting its robustness against multicollinearity and its superior efficiency. While SRDKE2 and SRDKE3 also outperform the LSE, ME, and DKE, they are slightly less effective than SRDKE1.

5. Real-Life Applications

The LSE, ME, DKE, and the suggested SRDKE1, SRDKE2, and SRDKE3 estimators’ performances via the MSE are investigated using two different real-life datasets.

5.1. Application 1

To confirm the superiority of the proposed estimators (SRDKE1, SRDKE2, and SRDKE3) over others, the real-life percentage data of development expenses and research in GNP of different countries (1972–1986) is used, in which these data are explained by Gruber [30], and Akdeniz and Erol [31]. The variables are as follows: y (US), ${\tilde{\mathit{x}}}_1$ (France), ${\tilde{\mathit{x}}}_2$ (West Germany), ${\tilde{\mathit{x}}}_3$ (Japan), ${\tilde{x}}_4$ (Soviet Union). The eigenvalues of $\tilde{S}$ are calculated as 302.9626, 0.7283, 0.0446, and 0.0345, and the condition number is 8781.5, indicating the multicollinearity occurrence. Hence, ${\widehat{\sigma }}^2$ is equal to 0.0015. Following Roozbeh et al. [27] and Roozbeh and Hamzah [28], the stochastic restriction is considered as $\tilde{R}=(1-2-2-2)$, where $\tilde{r}=\tilde{R}\beta +e$, $e~N(0,{\sigma }_{LSE}^2)$.

Table 11. The coefficients and MSEs.

	Existing Estimators			Proposed Estimators
Coef	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
${\alpha }_1$	0.6455	0.6095	0.6731	0.6349	0.6725	0.6725
${\alpha }_2$	0.0896	0.0990	0.0821	0.0919	0.0818	0.0816
${\alpha }_3$	0.1436	0.1579	0.1320	0.1475	0.1316	0.1318
${\alpha }_4$	0.1526	0.1567	0.1500	0.1543	0.1507	0.1508
$MSE$	0.0808	0.0686	0.0454	0.0389	0.0374	0.0374
$k$	-----	0.0014	-----	0.0014	0.0014	0.0014
$d$	-----	0.3444	-----	0.3444	0.3444	0.3444

mentions the coefficients and MSE of the estimators.

Table 11 clarifies the following comments:

• The LSE exhibits the worst performance as it has the highest MSE value.
• Both the ME and DKE demonstrate better performance than the LSE, with the ME outperforming the DKE.
• All the proposed estimators (SRDKE1, SRDKE2, and SRDKE3) have lower MSE values compared to the existing estimators, indicating that the proposed estimators outperform the existing ones.
• Among all the estimators, SRDKE2 and SRDKE3 achieve the best performance, followed by SRDKE1.

5.2. Application 2

Here, we use the popular real-life Hald data from Woods et al. [32]. These data are used and explained by many authors, including Kaçıranlar et al. [6], Lukman et al. [33], Kibria and Lukman [3], and Dawoud and Kibria [4], among others. The eigenvalues of $\tilde{S}$ are 44,676.206, $5965.422$, $809.952$, $105.419$, and the condition number is 425.3619. This indicates that severe multicollinearity exists.

We consider $\tilde{R}=(1-110)$ following Roozbeh et al. [27] and Roozbeh and Hamzah [28].

Table 12. The coefficients and MSEs.

	Existing Estimators			Proposed Estimators
Coef.	LSE	DKE	ME	SRDKE1	SRDKE2	SRDKE3
${\alpha }_1$	2.1930	2.1763	2.1818	2.1652	2.1654	2.1654
${\alpha }_2$	1.1533	1.1572	1.1562	1.1601	1.1600	1.1600
${\alpha }_3$	0.7585	0.7465	0.7490	0.7370	0.7372	0.7373
${\alpha }_4$	0.4863	0.4888	0.4882	0.4907	0.4906	0.4906
$MSE$	0.0638	0.0629	0.0625	0.0617	0.0617	0.0617
$k$	-----	0.3357	-----	0.3357	0.3357	0.3357
$d$	-----	0.8101	-----	0.8101	0.8101	0.8101

mentions the coefficients and MSE of estimators.

Table 12 clarifies the following comments:

• The LSE exhibits the worst performance as it has the highest MSE value.
• Both the ME and DKE demonstrate better performance than the LSE, with the ME outperforming the DKE.
• All the proposed estimators (SRDKE1, SRDKE2, and SRDKE3) have lower MSE values compared to the existing estimators, indicating that the proposed estimators outperform the existing ones.
• Among all estimators, the best performance is achieved by the three proposed estimators (SRDKE1, SRDKE2, and SRDKE3).

6. Conclusions

In this article, we propose three stochastic restricted biased estimators for the linear regression model. The properties of these proposed estimators are thoroughly discussed. Additionally, we derive the necessary and sufficient conditions under which the proposed estimators outperform the LSE, the ME, and the DKE, introduced by Dawoud and Kibria [4] to address multicollinearity. These conditions are established using the MSEM criterion, allowing for a comprehensive comparison of the estimators’ performance. To validate the theoretical findings, we conduct a simulation study, which demonstrates the superiority of the proposed estimators over the existing estimators. Furthermore, two real-life datasets are analyzed to illustrate the practical effectiveness of the proposed estimators. These new estimators can be extended to other models in future research. We recommend using the proposed stochastic restricted estimators in the presence of multicollinearity, with a careful selection of the tuning parameters $k$ and $d$ to achieve optimal performance.