Almost unbiased modified ridge-type estimator: An application to tourism sector data in Egypt

Abstract

This paper introduces an almost unbiased modified ridge-type estimator (AUMRTE) to avoid problems arising from multicollinearity. This estimator has the important features of the two important shrinkage estimators, the modified ridge-type estimator (MRTE) and almost unbiased estimator (AUE). We investigated the theoretical excellence of the proposed estimator according to the mean square error (MSE). We found that it has the superiority than the (MRTE) and almost unbiased two-parameter estimator (AUTE). Moreover, we run the simulation study, which depended on the simulated MSE (SMSE), squared bias (SB) and generalized cross-validation (GCV) as criteria to compare the estimators. The simulation results showed that the proposed estimator has the superiority than the estimators under comparison at several factors and at the same time, it works well at the high level of correlation. In addition, we investigated the behavior of the present estimator applying the real data. Under this trend, we applied the estimator to the tourism sector data in Egypt, which the results were consistent with the theoretical results.

Multicollinearity; Liu-type estimator; Ridge-type estimator; Almost unbiased modified ridge-type estimator

1. Introduction

The multicollinearity appears when the explanatory variables have high correlated. This problem has bad effect on the ordinary least squares (OLS) estimator, since it makes (MSE) is high and the estimator becomes unstable Yalian and Hu (2012). The ridge estimator (RE) which introduced by Hoerl and Kennard (1970) is stable solution to address the multicollinearity. This estimator is biased and it's adding more information to overcome the ill condition for the $\left(X^{T}X\right)$ matrix. In the same context, Liu (1993) proposed the biased estimator that called Liu estimator (LE) that mingling the stein estimator that introduced by Stein (1956) and (RE). For high level of multicollinearity the matrix $\left(X^{T}X\right)$ safer from ill condition with large condition number. The small value of ridge parameter cannot reduce the condition number by enough to overcome the ill condition. So that, Liu (2003) introduced Liu_type estimator (LTE) that depended on two parameters make together to reduce the condition number and at the same time improve the fitting and properties of the estimator Zhai et al. (2020). Ozkale and Kaciranlar (2007) suggest two-parameter estimator (TE), which has many features, since it contains the OLS, RE, Liu estimators in private situations. In fact, the (RE) and (LE) depend on OLS estimator, so we can use them in the case of low level of multicollinearity. Otherwise, the (TE) and (LTE) depend on any estimator, whether OLS or any other estimator. So that, we can use it at any level of multicolinearity. Sakallıoglu and Kaçıranlar (2008) and Yang and Chang (2010) modify the (LTE) in which it depends on (RE). This biased estimator has superior efficient than (RE), (LT) and (LTE). Furthermore, Omara (2019) modify the (TE) in which it depends on (RE). In addition, Yang and Chang (2010) develop a two-parameter estimator. Aslam and Ahmed (2020) suggested the class of biased estimator modify two-parameter estimator. Dorugade (2014) introduced the new biased estimator called ridge-type estimator (RTE). Lukman et al. (2019) modified the ridge-type estimator and proposed the new biased estimator called modify ridge-type estimator (MRTE). At the same time, Lukman et al. (2019) modified the ridge-type estimator with new prior information.

On the other hand, many studies go to minimize the estimators bias and at the same time keeping the MSE small. The almost unbiased estimator is one of important biased estimator which used to reduce the biased for the shrinkage estimators. There is a continuous need to improve the almost unbiased estimator to overcome the multicolinearity. In this direction, the statistical literature goes to improve the almost unbiased estimator performance by replacing the OLS estimator with more efficient shrinkage estimators. In this context, Singh et al. (1986a, 1986b) suggested almost unbiased ridge estimator. Furthermore, Singh et al. (1986a, 1986b) presented almost unbiased Liu estimator. In addition, Akdeniz and Kaciranlar (1995) suggested almost unbiased generalized Liu estimator (AUGLE). More on almost unbiased estimators, we refer our readers to Alheety and Kibria (2009), Alheety et al. (2021), Algamal (2021), Al-Taweel and Algamal (2020) and Al-Taweel and Algamal (2022). This study suggests new almost unbiased shrinkage estimator which we called almost unbiased modified ridge-type estimator (AUMRTE). This estimator merges the almost unbiased Liu estimator (AULE) with (MRTE).

The planning of the study is as follows: In section 2, we illustrated the model (subsection 2.1), proposed the estimators (subsection 2.2), and the provided the biasing parameters (subsection 2.3). The performance of the estimators according to the MSE is illustrated in section 3. The simulation study and its results are given in section 4. A real data are analyzed in section 5. Finally, the concluding are put in section 6.

2. Methodology

2.1. The regression model and shrinkage estimators

Conceder the linear regression model:

$$Y=X\beta +\mu$$ (1)

Where Y is $n\times 1$ is vector of response regression, X is $n\times p$ explanatory matrix, β is $p\times 1$ vector of coefficients and μ is $n\times 1$ vector of error term which follow a normal distribution such that $\mu \sim N(0,{\sigma }^2{I}_n)$. We simplify the linear regression model by use the canonical form. Since the matrix $\left(X^{T}X\right)$ is symmetric matrix, then $\mathrm{\Lambda }=Z^{T}Z=H{X}^{T}X{H}^T=\mathit{diag}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_p)$ where H and ${\lambda }_i$, $i=1,2,\dots ,p$ are the eigenvector and eigenvalues of the $\left(X^{T}X\right)$ matrix. The canonical model for the model in Eq. (1) is:

$$Y=Z\gamma +\mu$$ (2)

where $Z=XH$ and $\gamma =H^T\beta$.

For Eq. (2), (Hoerl and Kennard (1970) suggested RE to overcome the multicolinearity which formed as:

$${\hat{\gamma }}_{RE}\left(k\right)={(\mathrm{\Lambda }+kI)}^{-1}{Z}^{T}Y,k>0=H_{RE}{\hat{\gamma }}_{OLS}$$

where k is ridge parameter, $H_{RE}={(\mathrm{\Lambda }+kI)}^{-1}\mathrm{\Lambda }$ and ${\hat{\gamma }}_{OLS}$ is OLS estimator.

To overcome the high level of multicolinearity, (Liu, 1993) suggested LE which formed as:

$${\hat{\gamma }}_{LE}\left(d\right)={(\mathrm{\Lambda }+I)}^{-1}(Z^{T}Y-d{\hat{\gamma }}_{OLS}),0<d<1=H_{LE}{\hat{\gamma }}_{OLS}$$

where d is the biasing parameter and $H_{LE}={(\mathrm{\Lambda }+I)}^{-1}(\mathbf{\Lambda }-dI)$.

To get a new biased estimator has more ability to deal with multicolinearity, (Liu, 2003) introduced the LTE as:

$${\hat{\gamma }}_{LTE}(k,d)={(\mathrm{\Lambda }+kI)}^{-1}(Z^{T}Y-d{b}^{⁎}),k>0,-\infty <d<\infty$$

where $b^{⁎}$ is the any estimator for β.

Ozkale and Kaciranlar (2007) suggested two-parameter estimator, which formed as:

$${\hat{\gamma }}_{TE}(K,d)={(\mathrm{\Lambda }+kI)}^{-1}(Z^{T}Y-kd{\hat{\gamma }}_{OLS}),0<d<1,=H_{TE}{\hat{\gamma }}_{OLS}$$

where $H_{TE}={(\mathrm{\Lambda }+kI)}^{-1}(\mathbf{\Lambda }-kdI)$.

Dorugade (2014) modified the RE and suggested RTE which defined as:

$${\hat{\gamma }}_{RTE}(k,d)={(\mathrm{\Lambda }+kdI)}^{-1}{Z}^{T}Y,k>0,0<d<1$$

Lukman et al. (2019) modified the RTE and suggested MRTE as:

$${\hat{\gamma }}_{MRTE}(k,d)={(\mathrm{\Lambda }+k(1+d)I)}^{-1}{Z}^{T}Y,k>0,0<d<1=H_{MRTE}{\hat{\gamma }}_{OLS}$$ (3)

where $H_{MRTE}={(\mathrm{\Lambda }+k(1+d)I)}^{-1}\mathrm{\Lambda }$

$$E\left({\hat{\gamma }}_{MRTE}(k,d)\right)=H_{MRTE}\mathit{\gamma }$$

$$bias\left({\hat{\gamma }}_{MRTE}(k,d)\right)=(H_{MRTE}-I)\mathit{\gamma }$$

$$Cov\left({\hat{\gamma }}_{MRTE}(k,d)\right)={\sigma }^2{H}_{MRTE}\mathrm{\Lambda }{H}_{MRTE}^T$$ (4)

$$\begin{gathered} MSE\left({\hat{\mathit{\gamma }}}_{MRTE}(k,d)\right)={\sigma }^2{H}_{MRTE}\mathrm{\Lambda }{H}_{MRTE}^T \\ +(H_{MRTE}-I){\mathit{\gamma }\mathit{\gamma }}^{\mathit{T}}{(H_{MRTE}-I)}^T \end{gathered}$$

$$MSE\left({\hat{\mathit{\gamma }}}_{MRTE}(k,d)\right)={\sigma }^2\sum _{i=1}^p\frac{{\lambda }_i}{{({\lambda }_i+k(1+d))}^2}+\sum _{i=1}^p\frac{{\mathit{\gamma }}_i^2}{{({\lambda }_i+k(1+d)-I)}^2}$$

Kadiyala (1984) suggested class of almost unbiased estimator (AUE) which defined as:

$${\hat{\gamma }}_{\mathrm{AU}E}=[I+{(\mathrm{\Lambda }+I)}^{-1}(1-d)]{\hat{\gamma }}_{OLS}0<d<1$$ (5)

There is a continuous need to improve the almost unbiased estimator which used to overcome the multicolinearity. In this direction, the statistical literature goes to improve the performance of these estimators by replacing the OLS estimator with more efficient punishing estimators. Akdeniz and Kaciranlar (1995) suggested almost unbiased generalized Liu estimator (AUGLE) which defined as:

$${\hat{\gamma }}_{\mathrm{AUL}E}=[I+(I-D){(\mathrm{\Lambda }+I)}^{-1}]{\hat{\gamma }}_{liu}0<d<1=[I-{(I-D)}^2{(\mathrm{\Lambda }+I)}^{-2}]{\hat{\gamma }}_{OLS}$$

where $D=dig(d_1,d_2,\dots ,d_p)$.

Wu and Young (2013) introduced almost unbiased two-parameter estimator (AUTE) as:

$${\hat{\gamma }}_{\mathrm{AUT}E}=[I+k(I-d){(\mathrm{\Lambda }+kI)}^{-1}]{\hat{\gamma }}_{liu}0<d<1=[I-k^2{(I-d)}^2{(\mathrm{\Lambda }+kI)}^{-2}]{\hat{\gamma }}_{OLS}=H_{\mathrm{AUT}E}{\hat{\gamma }}_{OLS}$$

where $H_{\mathrm{AUT}E}=[I-k^2{(I-d)}^2{(\mathrm{\Lambda }+kI)}^{-2}]$

$$E\left({\hat{\gamma }}_{\mathrm{AUT}E}(k,d)\right)=H_{\mathrm{AUT}E}\mathit{\gamma }$$

$$bias\left({\hat{\gamma }}_{\mathrm{AUT}E}(k,d)\right)=(H_{\mathrm{AUT}E}-I)\mathit{\gamma }$$

$$Cov\left({\hat{\gamma }}_{\mathrm{AUT}E}(k,d)\right)={\sigma }^2{H}_{\mathrm{AUT}E}{\mathrm{\Lambda }}^{-1}{H}_{\mathrm{AUL}E}$$ (6)

$$\begin{gathered} MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUT}E}(k,d)\right)={\sigma }^2{H}_{\mathrm{AUT}E}{\mathrm{\Lambda }}^{-1}{H}_{\mathrm{AUT}E} \\ +(H_{\mathrm{AUT}E}-I){\mathit{\gamma }\mathit{\gamma }}^{\mathit{T}}{(H_{\mathrm{AUT}E}-I)}^T \end{gathered}$$

$$\begin{gathered} MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUT}E}(k,d)\right)={\sigma }^2\sum _{i=1}^p\frac{1}{{\lambda }_i}{(1-\frac{k^2{(1-d)}^2}{{({\lambda }_i+kI)}^2})}^2 \\ +\sum _{i=1}^p{\left(\frac{k^2{(1-d)}^2}{{({\lambda }_i+kI)}^2}\right)}^2{\mathit{\gamma }}_i^2 \end{gathered}$$

2.2. The proposed estimator:

In begging, we illustrated the (Xu and Yang, 2011) definition of almost unbiased estimator.

Definition

Let $\hat{\beta }$ be a biased estimator of the β such that $Bias\left(\hat{\beta }\right)=E\left(\hat{\beta }\right)-\beta =H\beta$ and $(\hat{\beta }-H\beta )=\beta$, then we defined the almost unbiased estimator as $\tilde{\beta }=\hat{\beta }-H\beta =(I-H)\hat{\beta }$.

Because of the preference of the (MRTE) estimator over the (RE), (LE) and (MRTE), we find that, this estimator is a candidate to merge with (AUE). We use Eq. (3) and Eq. (5) then defined this new biased estimator (AUMRTE) as:

$${\hat{\gamma }}_{\mathrm{AUMRTE}}(k,d)=[I-({(\mathrm{\Lambda }+k(1+d)I)}^{-1}\mathrm{\Lambda }-I)]{\hat{\gamma }}_{MRTE}k>0,0<d<1=[2I-{(\mathrm{\Lambda }+k(1+d)I)}^{-1}\mathrm{\Lambda }]{(\mathrm{\Lambda }+k(1+d)I)}^{-1}\mathrm{\Lambda }{\hat{\gamma }}_{OLS}=[I-k^2{(1+d)}^2{(\mathrm{\Lambda }+k(1+d)I)}^{-2}]{\hat{\gamma }}_{OLS}=H_{\mathrm{AUMRTE}}{\hat{\gamma }}_{OLS}$$

The expected, biased and MSE for the new estimator is formed as:

$$E\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)=H_{\mathrm{AUMRTE}}\mathit{\gamma }$$

$$bias\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)=(H_{\mathrm{AUMRTE}}-I)\mathit{\gamma }$$

$$Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)={\sigma }^2{H}_{\mathrm{AUMRTE}}{\mathrm{\Lambda }}^{-1}{H}_{\mathrm{AUMRTE}}$$ (7)

$$\begin{gathered} MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)={\sigma }^2{H}_{\mathrm{AUMRTE}}{\mathrm{\Lambda }}^{-1}{H}_{\mathrm{AUMRTE}} \\ +(H_{\mathrm{AUMRTE}}-I){\mathit{\gamma }\mathit{\gamma }}^{\mathit{T}}{(H_{\mathrm{AUMRTE}}-I)}^T={\sigma }^2\sum _{i=1}^p\frac{1}{{\lambda }_i}{(1-\frac{k^2{(1+d)}^2}{{({\lambda }_i+k(1+d))}^2})}^2 \\ +\sum _{i=1}^p{\left(\frac{k^2{(1+d)}^2}{{({\lambda }_i+k(1+d))}^2}\right)}^2{\gamma }_i^2 \end{gathered}$$ (8)

The AUMRTE estimator has some special case that we can follow as:

• a.If $k=0$ and $d=0$ then ${\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(0,0)={\hat{\mathit{\gamma }}}_{OLS}$.
• b.If $d=1$ and $k=\frac{k}{2}$ then ${\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(\frac{k}{2},1)={\hat{\mathit{\gamma }}}_{\mathrm{AURE}}\left(k\right)$.
• c.If $d+1=d$, then ${\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)={\hat{\mathit{\gamma }}}_{\mathrm{AUTE}}\left(k\right)$.

2.3. Choosing the shrinkage parameters ($k,d$)

For the proposed estimator, we determine the chosen method for the shrinkage parameters $k,d$. Let $e=1+d$ and $MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)={\sigma }^2{\sum }_{i=1}^p\frac{1}{{\lambda }_i}{(1-{\mathit{v}}_{\mathit{i}}^2)}^2+{\sum }_{i=1}^p{\mathit{v}}_{\mathit{i}}^4{\gamma }_i^2$ where ${\mathit{v}}_{\mathit{i}}=k(1+d)/({\lambda }_i+k(1+d))=ke/({\lambda }_i+ke)$. To get the optimal value of d, let k fixed and take the derivative of MSE in Eq. (8) with respect to d, then equalize the result to zero.

$$\frac{\partial MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)}{\partial \mathit{d}}=\frac{\partial MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)}{\partial {\mathit{v}}_{\mathit{i}}}\frac{\partial {\mathit{v}}_{\mathit{i}}}{\partial \mathit{d}}=0$$

Since $\partial {\mathit{v}}_{\mathit{i}}/\partial \mathit{d}=k{\lambda }_i/{({\lambda }_i+k(1+d))}^2\ne 0$, then

$$\begin{gathered} \frac{\partial MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)}{\partial {\mathit{v}}_{\mathit{i}}}=-4{\sigma }^2\sum _{i=1}^p\frac{{\mathit{v}}_{\mathit{i}}(1-{\mathit{v}}_{\mathit{i}}^2)}{{\lambda }_i}+4\sum _{i=1}^p{\mathit{v}}_{\mathit{i}}^{\mathbf{3}}{\gamma }_i^2=0 \\ -{\sigma }^2\sum _{i=1}^p(1-{\mathit{v}}_{\mathit{i}}^2)+\sum _{i=1}^p{\lambda }_i{\mathit{v}}_{\mathit{i}}^{\mathbf{2}}{\gamma }_i^2=0 \\ -{\sigma }^2+{\sigma }^2{\mathit{v}}_{\mathit{i}}^2+{\lambda }_i{\mathit{v}}_{\mathit{i}}^{\mathbf{2}}{\gamma }_i^2=0 \end{gathered}$$ (9)

Since ${\mathit{v}}_{\mathit{i}}=ke/({\lambda }_i+ke)$, then for Eq. (9), we find

$$e_{opt}=\frac{{\lambda }_i\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}}}{k(1-\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}})}$$

and

$$d_{opt}=\frac{{\lambda }_i\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}}-k(1-\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}})}{k(\mathbf{1}-\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}})}$$ (10)

For workable applied purpose, ${\sigma }^2$ and ${\gamma }_i^2$ are replaced with ${\hat{\sigma }}^2$ and ${\hat{\gamma }}_i^2$. Then the Eq. (10) become

$$d_{opt}=\frac{{\lambda }_i\sqrt{\frac{{\hat{\sigma }}^2}{{\hat{\sigma }}^2+{\lambda }_i{\hat{\gamma }}_i^2}}-k(1-\sqrt{\frac{{\hat{\sigma }}^2}{{\hat{\sigma }}^2+{\lambda }_i{\hat{\gamma }}_i^2}})}{k(1-\sqrt{\frac{{\hat{\sigma }}^2}{{\hat{\sigma }}^2+{\lambda }_i{\hat{\gamma }}_i^2}})}$$ (11)

To get the optimal value of k, let d fixed and take the derivative of MSE in Eq. (8) with respect to the parameter k, then equalize the result to zero.

$$\frac{\partial MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)}{\partial \mathit{k}}=\frac{\partial MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)}{\partial {\mathit{v}}_{\mathit{i}}}\frac{\partial {\mathit{v}}_{\mathit{i}}}{\partial \mathit{k}}=0$$

Since $\partial {\mathit{v}}_{\mathit{i}}/\partial \mathit{k}={\lambda }_i(1+d)/{({\lambda }_i+k(1+d))}^2\ne 0$, then for Eq. (9), we get

$$k_{opt}=\frac{{\lambda }_i\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}}}{(d+1)(1-\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}})}$$ (12)

If we follow the Kibria (2003) which proposed the arithmetic means of k value, then we can propose the arithmetic means of $k_{opt}$ as

$$k_{opt}=\frac{1}{p}\sum _{i=1}^p\frac{{\lambda }_i\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}}}{(d+1)(1-\sqrt{\frac{{\sigma }^2}{{\sigma }^2+{\lambda }_i{\gamma }_i^2}})}$$ (13)

For workable applied purpose, ${\sigma }^2$ and ${\gamma }_i^2$ are replaced with ${\hat{\sigma }}^2$ and ${\hat{\gamma }}_i^2$. Then the Eq. (13) becomes

$$k_{opt}=\frac{1}{p}\sum _{i=1}^p\frac{{\lambda }_i\sqrt{\frac{{\hat{\sigma }}^2}{{\hat{\sigma }}^2+{\lambda }_i{\hat{\gamma }}_i^2}}}{(d+1)(1-\sqrt{\frac{{\hat{\sigma }}^2}{{\hat{\sigma }}^2+{\lambda }_i{\hat{\gamma }}_i^2}})}$$ (14)

One of important methods that used to obtain the optimal shrinkage parameters k and d, is a generalized cross-validation (GCV). This method makes a equilibrium between the estimator's prediction accuracy and the bias which caused by the shrinkage the estimator (Arashi et al., 2021). The GCV received attention in the statistical literature, (Omara, 2019) use GCV to choose the shrinkage parameters for two-parameter ridge estimator. Moreover, (Roozbeh et al., 2020) use GCV as a criterion to compare the estimators for ridge rank regression. In fact, the GCV has good properties and at the same time it has a simplicity. We can choose the parameters for the propose estimator ${\mathit{k}}_{\mathit{G}\mathit{C}\mathit{V}}$ and $d_{GCV}$ by minimizing the following unobservable risk function which defined as

$$R_{\mathrm{AUMRTE}}(d,k)=\frac{1}{n}{(y-\hat{y})}^T(y-\hat{y})$$

where $\hat{y}=Z{\hat{\gamma }}_{\mathrm{AUMRTE}}(d,k)=L(d,k)Y$ and $L(d,k)=Z[I-k^2{(1+d)}^2{(\mathrm{\Lambda }+k(1+d)I)}^{-2}]{\left({\mathrm{Z}}^T\mathrm{Z}\right)}^{-1}{Z}^T$ which can be defined as almost unbiased modified ridge-type hat matrix of y. Then the $GCV(k.d)$ defined as

$$GCV(k.d)=\frac{{\Vert y-Z{\hat{\gamma }}_{\mathrm{AUMRTE}}(k.d)\Vert }^2}{{(I-n^{-1}tr\left(L(k,d)\right))}^2}=\frac{{\Vert (I-L(k,d))y\Vert }^2}{{(I-n^{-1}tr\left(L(k,d)\right))}^2}$$

There are various methods exist to estimate shrinkage parameters. To mention a few, see Suhail and Kibria (2021) and Lukman et al. (2019) among others.

3. The MSE comparison

In this section, we make a comparison between AUTE and MRTE according to the (MSE). To check the superiority of the proposed estimator, we need the following lemma:

Lemma 1

Let C be a positive definite $(p.d)$ matrix and c be a vector, then $C-c{c}^T\ge 0$ , iff $c^T{C}^{-1}c\le 1$ . See Farebrother (1976) .

Lemma 2

Let ${\hat{\beta }}_{\mathrm{i}}=H_{i}y$ , $i=1,2$ be two estimators of β. Assume that $\mathrm{\nabla }=Cov\left({\hat{\beta }}_1\right)-Cov\left({\hat{\beta }}_2\right)$ be a $p.d$ , then $MSE\left({\hat{\beta }}_1\right)-MSE\left({\hat{\beta }}_2\right)\ge 0$ , iff ${(H_2-I)}^T{(\mathrm{\nabla }+{(H_1-I)}^T{(H_1-I)}^T)}^{-1}(H_2-I)\le 1$ . See Trenkler and Toutenburg, 1990 .

Theorem 1

Consider ${\hat{\mathit{\gamma }}}_{\mathrm{AUTE}}(k,d)$ and ${\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)$ are two estimators of β. Assume that ${\mathrm{\nabla }}_1=Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUTE}}(k,d)\right)-Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)\mathit{\text{be a}}(p.d)$ , then $MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUT}E}(k,d)\right)-MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)\ge 0$ , iff $(H_{\mathrm{AUMRTE}}-I_n){({\mathrm{\nabla }}_1+{(H_{\mathrm{AUTE}}-I)}^T{(H_{\mathrm{AUTE}}-I)}^T)}^{-1}{(H_{\mathrm{AUMRTE}}-I_n)}^T\le 1$ .

Proof

From Eq. (6) and (7), we find that:

$${\mathrm{\nabla }}_1=Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUTE}}(k,d)\right)-Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)={\sigma }^2\left[\sum _{i=1}^p\frac{1}{{\lambda }_i}[{(1-\frac{k^2{(1-d)}^2}{{({\lambda }_i+k)}^2})}^2-{(1-\frac{k^2{(1+d)}^2}{{({\lambda }_i+k(1+d))}^2})}^2]\right]$$

This inequality is $(p.d)$ if and only if

$$\frac{k^2{(1-d)}^2}{{({\lambda }_i+k)}^2}-\frac{k^2{(1+d)}^2}{{({\lambda }_i+k(1+d))}^2}<0$$

$$\frac{{(1-d)}^2{({\lambda }_i+k(1+d))}^2-{(1+d)}^2{({\lambda }_i+k)}^2}{{({\lambda }_i+k)}^2{({\lambda }_i+k(1+d))}^2}<0$$

This inequality is true if and only if

$${(1-d)}^2{({\lambda }_i+k(1+d))}^2-{(1+d)}^2{({\lambda }_i+k)}^2<0$$

This inequality is true for $k>0$ and $0<d<1$.

By using Lemma 2, the proof completed.

Theorem 2

Consider ${\hat{\gamma }}_{MRTE}(k,d)\mathit{\text{and}}{\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)$ are two estimators of β. Assume that ${\mathrm{\nabla }}_2=Cov\left({\hat{\gamma }}_{MRTE}(k,d)\right)-Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)\mathit{\text{be a}}(p.d)$ , then $MSE\left({\hat{\gamma }}_{MRTE}(k,d)\right)-MSE\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)\ge 0$ , iff

$$\begin{gathered} (H_{\mathrm{AUMRTE}}-I_n){({\mathrm{\nabla }}_2+{(H_{MRTE}-I)}^T{(H_{MRTE}-I)}^T)}^{-1} \\ \times {(H_{\mathrm{AUMRTE}}-I_n)}^T\le 1. \end{gathered}$$

Proof

From Eq. (4) and (7), we find that:

$${\mathrm{\nabla }}_2=Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{MRT}E}(k,d)\right)-Cov\left({\hat{\mathit{\gamma }}}_{\mathrm{AUMRTE}}(k,d)\right)={\sigma }^2\sum _{i=1}^p[{\left(\frac{{\lambda }_i}{{({\lambda }_i+k(1+d))}^2}\right)}^2-\frac{1}{{\lambda }_i}{(1-\frac{k^2{(1+d)}^2}{{({\lambda }_i+k(1+d))}^2})}^2]={\sigma }^2\sum _{i=1}^p[{\left(\frac{{\lambda }_i}{{({\lambda }_i+k(1+d))}^2}\right)}^2-\frac{1}{{\lambda }_i}{\left(\frac{{({\lambda }_i+k(1+d))}^2-k^2{(1+d)}^2}{{({\lambda }_i+k(1+d))}^2}\right)}^2]={\sigma }^2\sum _{i=1}^p\left[\frac{{\lambda }_i^3{({\lambda }_i+k(1+d))}^4-{({({\lambda }_i+k(1+d))}^2-k^2{(1+d)}^2)}^2}{{\lambda }_i{({\lambda }_i+k(1+d))}^4}\right]$$

This inequality is true if and only if

$${\lambda }_i^3{({\lambda }_i+k(1+d))}^4-{({({\lambda }_i+k(1+d))}^2-k^2{(1+d)}^2)}^2>0$$

$$\begin{gathered} {\lambda }_i^3{({\lambda }_i+k(1+d))}^4-{({\lambda }_i+k(1+d))}^4 \\ +2k^2{({\lambda }_i+k(1+d))}^2{(1+d)}^2-k^4{(1+d)}^4>0 \end{gathered}$$

$$({\lambda }_i^3-1){({\lambda }_i+k(1+d))}^4-k^4{(1+d)}^4+2k^2{({\lambda }_i+k(1+d))}^2{(1+d)}^2>0$$

This inequality is true for $0<d<1$ and $k>0$, then according to Lemma 2, the proof completed.

4. Simulation study

To verify the effectiveness of the proposed estimator comparing with the other estimators according to a set of factors, we run the following simulation study. The comparisons are made between the suggested estimator and the MRTE and AUTE. All these comparisons are performed by the Matlab Programming. On the beginning, we use the following model to determine dependent variable data:

$$y_i=\sum _{j=1}^p{x}_{ij}{\beta }_i+{\mu }_i,i=1,2,\dots ,n$$

where $\mu \sim iidN(0,{\sigma }^2)$.

There are many factors are candidate in this simulation. The levels of correlation between the explanatory variables are important factor for comparing the variables. To ensure that there are degrees of this correlation between the explanatory variables, we follow McDonald and Galarneau (1975), and generates the explanatory variables by

$$x_{ij}={(1-{\rho }^2)}^{1/2}{\omega }_{ij}+\rho {\omega }_{i,p+1},i=1,2,\dots ,n,j=1,2,\dots ,p$$

where ${\omega }_{ij}$, s are pseudo-random numbers such that $\omega \sim iidN(0,1)$ and ρ is the correlation between the explanatory variables. To show the effect of the levels of correlation between the explanatory variables at the different levels, we choose this correlation as $\rho =0.90,0.95,0.99$. Moreover, we use the three levels of sample size $n=50,100$ and 200. We take the number of explanatory $p=5$ and 30. We take the variance of the error term as ${\sigma }^2=0.1$ and 20. In addition, the simulation is repeated 2000 times and we use the simulated MSE (SMSE) and squared bias (SB) as a criterion to compare between the estimators, such that they are formed at Eq. (15) and Eq. (16). In addition, in order to avoid the over fitting, we use the (GCV) as a reliable criterion.

$$SMSE\left(\hat{\beta }\right)=\frac{{\sum }_{r=1}^{2000}{({\hat{\beta }}_r-\beta )}^T({\hat{\beta }}_r-\beta )}{2000}$$ (15)

$$SB\left(\hat{\beta }\right)={(\frac{{\sum }_{r=1}^{2000}{\hat{\beta }}_r^T}{2000}-\beta )}^T(\frac{{\sum }_{r=1}^{2000}{\hat{\beta }}_r^T}{2000}-\beta )$$ (16)

where ${\hat{\beta }}_r$ is the estimator in $r^{th}$ iteration of the simulation. The initial value for β is selected as ${\beta }_{p=5}={[2,2,2,2,2]}^T$ and ${\beta }_{p=30}={[2,2,\dots ,2]}^T$. To select the value of k and d we use the follow the plan:

• ■For AUMRTE, we use the Eq. (11) to Eq. (14).
• ■
  For MRTE, we follow Lukman et al. (2019) and use $d_{opt}\left(\mathrm{MRTE}\right)=\left(\frac{{\hat{\sigma }}^2}{k{\gamma }_i^2}\right)-1$ and the arithmetic means of $k_{opt}\left(\mathrm{MRTE}\right)$ as

  $$k_{opt}\left(\mathrm{MRTE}\right)=\frac{1}{p}\sum _{i=1}^p\frac{{\hat{\sigma }}^2}{(1+d){\hat{\gamma }}_i^2}$$
• ■
  For AUTE, we follow Wu and Young (2013) and use $d_{opt}\left(\mathrm{AUTE}\right)=1-\frac{({\lambda }_i+k)\sigma }{k}\sqrt{\frac{1}{{\sigma }^2-{\lambda }_i{\gamma }_i^2}}$ and the arithmetic means of $k_{opt}\left(\mathrm{AUTE}\right)$ as

  $$k_{opt}\left(\mathrm{AUTE}\right)=\frac{1}{p}\sum _{i=1}^p\frac{\sigma {\lambda }_i}{(1+d)\sqrt{{\sigma }^2+{\lambda }_i{\gamma }_i^2}-\sigma }$$

To determine the priority of the selected estimators, the criteria SMSE, SB and GCV were used. We summarized the simulation results according to these criteria in Table 1, Table 2, Table 3. The results show that, the AUMRTE has the smallest SMSE, SB and GCV at all the factors. In contrast, the MRTE got the largest SMSE and SB at the all factors. Additionally, at $p=5$, AUMRTE performs well at $\rho =0.95$ compared to the other correlation levels but at $p=30$, it performs well at $\rho =0.90$ compared to the other correlation levels. Moreover, at all estimators, the SMSE, SB and GCV are tended to increase with increasing of the variance of the error and the number of explanatory variables and at the same time it tends to decrease with increasing of sample size. It is clear that, at the most case there is agreement between the results of SMSE, SB and GCV. According to GCV, we find that there is a greater positive effect of increasing the size of sample and decreasing the number of explanatory variable on the work of estimators. When the correlation be trend to high, the efficiency of the estimators decreases, and it is worse at $\rho =0.99$.

Table 1.

The SMSE value for the estimators at deferent factors.

p	n	ρ	${\sigma }^2=0.1$			${\sigma }^2=20$
			MRTE	AUTE	AUMRTE	MRTE	AUTE	AUMRTE
5	50	0.90	0.0212	0.0145	0.0088	0.0951	0.0882	0.0521
		0.95	0.0135	0.0124	0.0064	0.0665	0.0616	0.0328
		0.99	0.0193	0.0149	0.0072	0.0813	0.0917	0.0339
	100	0.90	0.0179	0.0118	0.0081	0.0733	0.0649	0.0491
		0.95	0.0179	0.0095	0.0053	0.0645	0.0592	0.0353
		0.99	0.0154	0.0115	0.0061	0.0684	0.0633	0.0399
	200	0.90	0.0081	0.0042	0.0039	0.0557	0.0412	0.0321
		0.95	0.0054	0.0088	0.0021	0.0293	0.0201	0.0119
		0.99	0.0028	0.0059	0.0017	0.0113	0.0092	0.0107
30	50	0.90	0.0508	0.0479	0.0286	0.1991	0.0941	0.0928
		0.95	0.0685	0.0622	0.0292	0.2012	0.0962	0.0999
		0.99	0.0717	0.0719	0.0305	0.2054	0.0998	0.1021
	100	0.90	0.0554	0.0451	0.0125	0.0912	0.0804	0.0664
		0.95	0.0691	0.0618	0.0141	0.1054	0.0921	0.0691
		0.99	0.0737	0.0792	0.064	0.1098	0.0978	0.0709
	200	0.90	0.0192	0.0149	0.0108	0.0628	0.0611	0.0492
		0.95	0.0204	0.0158	0.0111	0.0707	0.0744	0.0709
		0.99	0.0211	0.0164	0.0122	0.0777	0.0791	0.0717

Table 2.

The $SB\left(\hat{\beta }\right)$ value for the estimators at deferent factors.

p	n	ρ	${\sigma }^2=0.1$			${\sigma }^2=20$
			MRTE	AUTE	AUMRTE	MRTE	AUTE	AUMRTE
5	50	0.90	0.0091	0.0073	0.0049	0.0228	0.0199	0.0147
		0.95	0.0071	0.0088	0.0041	0.0199	0.0228	0.0318
		0.99	0.0082	0.0064	0.0047	0.0208	0.0182	0.0117
	100	0.90	0.0066	0.0052	0.0033	0.0097	0.0081	0.0048
		095	0.0039	0.0021	0.0028	0.0066	0.0072	0.0029
		0.99	0.0048	0.0048	0.0031	0.0077	0.0081	0.0047
	200	0.90	0.0061	0.0047	0.0025	0.0114	0.0097	0.0111
		095	0.0033	0.0011	0.0011	0.0098	0.0087	0.0098
		0.99	0.0044	0.0021	0.0021	0.0109	0.0094	0.0104
30	50	0.90	0.0108	0.0099	0.0082	0.0335	0.0333	0.0301
		0.95	0.0111	0.0102	0.0088	0.0369	0.0399	0.0304
		0.99	0.0124	0.0105	0.0099	0.0392	0.0431	0.0391
	100	0.90	0.0101	0.0094	0.0088	0.0111	0.0105	0.0101
		0.95	0.0118	0.0091	0.0074	0.0109	0.0103	0.0098
		0.99	0.0147	0.0092	0.0073	0.0108	0.0104	0.0097
	200	0.90	0.0092	0.0081	0.0066	0.0108	0.0102	0.0092
		0.95	0.0099	0.0084	0.0077	0.0111	0.0112	0.0098
		0.99	0.0121	0.0089	0.0081	0.0116	0.0131	0.0103

Table 3.

The $GCV\left(\hat{\beta }\right)$ value for the estimators at deferent factors.

p	n	ρ	${\sigma }^2=0.1$			${\sigma }^2=20$
			MRTE	AUTE	AUMRTE	MRTE	AUTE	AUMRTE
5	50	0.90	6.3652	5.0891	1.0012	6.6054	5.3065	1.0932
		0.95	6.5426	5.1325	1.0156	6.7742	5.5243	1.1318
		0.99	6.7082	5.4089	1.0827	6.9271	5.6284	1.3084
	100	0.90	4.0962	3.0514	0.9564	5.1802	2.9905	0.9654
		095	4.1564	3.0328	1.0015	5.3652	2.8638	0.9912
		0.99	4.2291	3.0632	1.0314	5.5021	2.4732	1.0214
	200	0.90	2.0185	1.9521	0.6965	4.6582	2.0658	0.9051
		095	2.0012	2.2518	0.8876	4.1068	2.6391	1.0325
		0.99	1.9834	2.3212	0.9765	4.0619	2.9653	1.0912
30	50	0.90	9.3255	8.9142	2.0879	9.9085	9.8256	2.7354
		0.95	9.7621	9.0698	2.1864	10.0304	9.9241	2.9941
		0.99	9.9025	9.3262	2.6152	10.2142	10.0251	3.2014
	100	0.90	7.6253	7.1982	1.9021	7.9056	7.3654	2.0214
		0.95	7.8641	7.2356	1.9214	8.0654	7.6021	2.3127
		0.99	7.9054	7.8321	1.9325	8.3712	8.0654	2.4085
	200	0.90	5.3242	4.8947	1.0245	5.9142	5.0214	1.0524
		0.95	5.5931	4.9921	1.0382	6.0325	5.1521	1.0934
		0.99	5.9342	5.0932	1.0634	6.1821	5.3921	1.1186

5. Application to the tourism sector data in Egypt

In this section, we confirm our results by application to the tourism data in Egypt (1995: 2019). We take the data form the Central Agency for Public Mobilization and Statistics (https://www.capmas.gov.eg/). We use the GDP of the tourism as dependent variable (y) and number of tourist nights (X₁), number of tourists (X₂) and total investments in the tourism (X₃) and number of workers in the tourism (X₄) as explanatory variables. We use the (SMSE), (SB), (R²) and GCV as a criteria's. Firstly, it is necessary to verify the existence of a multicollinearity between the independent variables and what is the level of it is. In this direction, we found that the correlation between the explanatory variables is between 0.684 and 0.893, which indicates the existence of a multicollinearity. To determine the level of multicollinearity, we use the condition number $CN=\sqrt{{\lambda }_{max}/{\lambda }_{min}}$, where ${\lambda }_{max}\text{and}{\lambda }_{min}$ are the largest and lowest eigenvalues for the $\left(X^{T}X\right)$ matrix. We find that, the eigenvalues of the $\left(X^{T}X\right)$ matrix equal 3151,435.65 and 2.94, then the $CN=32.73$. This result means that there is strong level of multicollinearity.

For choosing the initial values of β, we follow the group of studies and articles related to the factors affecting tourism in Egypt were followed, including (Abd El-hamed (2021), Elawa (2014), Dibas (2001)), through which it was concluded that the approximate average of the effect of a unit increase of both the number of tourist nights, the number of tourist, total investments in the tourism sector and the number of workers in the tourism sector are 0.521, 0.607, 0.318 and 0.154. So we can suggest the initial value of β as $\beta ={[0.105,0.521,0.607,0.318,0.154]}^T$. We summarize the results of estimate the model and the value of (SMSE), (SB), (R²) and GCV for the estimators at Table 4.

Table 4.

The coefficients of estimators and the value of (SMSE), (SB), R² and GCV.

Coefficients	Case (1)			Case (2)			Case (3)
	${\mathit{k}}_{\mathit{o}\mathit{p}\mathit{t}},{\mathit{d}}_{\mathit{o}\mathit{p}\mathit{t}}$			${\mathit{k}}_{\mathit{G}\mathit{C}\mathit{V}},{\mathit{d}}_{\mathit{o}\mathit{p}\mathit{t}}$			${\mathit{k}}_{\mathit{o}\mathit{p}\mathit{t}},{\mathit{d}}_{\mathit{G}\mathit{C}\mathit{V}}$
	MRTE	AUTE	AUMRTE	MRTE	AUTE	AUMRTE	MRTE	AUTE	AUMRTE
β0	0.315	0.357	0.328	0.220	0.251	0.204	0.224	0.238	0.241
β1	0.782	0.711	0.702	0.547	0.557	0.528	0.664	0.608	0.611
β2	0.682	0.642	0.637	0.604	0.614	0.635	0.589	0.599	0.582
β3	0.405	0.425	0.435	0.339	0.358	0.445	0.507	0.514	0.504
β4	0.152	0.195	0.138	0.204	0.228	0.237	0.243	0.251	0.248
MSE	0.054	0.068	0.0082	0.011	0.0024	0.00057	0.0092	0.00097	8.24 × 10−4
SB	0.047	0.051	0.0057	0.0087	0.00091	1.34 × 10−4	0.0075	1.94 × 10−4	1.32 × 10−4
GCV	12.814	12.325	11.815	12.581	12.183	11.708	11.732	11.5514	11.2913
R2	0.6647	0.6624	0.6831	0.6845	0.7154	0.7408	0.6925	0.7225	0.7584

The results in Table 4 indicate the sign of the coefficients are identical for all estimators. In addition, the results of coefficient of determination ${\mathit{R}}^{\mathbf{2}}$ of AUMRTE reach to 0.6831, 0.7408 and 0.7584 at cases (1-3) for Table 4 which is greater than the other estimators. These results indicate that the AUMTE improve the prediction accuracy for the model. In addition, for all cases, the AUMRTE and MRTE have the lowest and largest value of SMSE, SB and GCV.

Fig. 1 shows the ratio of the observation to the expected value against the years. At AUMRTE, the ratios at all years are close to the line of 100%, which indicate that the AUMRTE gives the model high predictive power. In the same way the AUTE give the model high predictive power than the MRTE. For get the good vision about the AUMRTE, we plot the GCV value ve. d and k for AUMRTE and presented in Fig. 2. This figure, shows three cases of optimal values of k and d that minimize the GCV's, these cases are (${\mathit{k}}_{\mathit{o}\mathit{p}\mathit{t}}$, ${\mathit{d}}_{\mathit{o}\mathit{p}\mathit{t}}$-${\mathit{k}}_{\mathit{G}\mathit{C}\mathit{V}}$, ${\mathit{d}}_{\mathit{o}\mathit{p}\mathit{t}}$-${\mathit{k}}_{\mathit{o}\mathit{p}\mathit{t}}$, ${\mathit{d}}_{\mathit{G}\mathit{C}\mathit{V}}$). For Fig. 2, the GCV's are convex functions and at the same time have a global minimum. In addition, it shown that, for AUMTE, the lowest value of GCV occurs in case (b), which it reached to 11.2913.

Figure 1.

Plots of the ratio of observation to expected value $(Y/\hat{Y})$ versus years.

Figure 2.

Plots of the GCV(k,d) value versus d and k for AUMRTE.

6. Conclusion

This paper suggested a new almost unbiased ridge-type estimator (AUMRTE) to deal with multicollinearity. Theoretical comparisons were made between the AUMRTE and each of MRTE and AUTE based on (MSE). These comparisons showed that the superiority of the AUMRTE over both MRTE and AUTE. The simulation study used SMSE, SB and GCV as criteria for comparison, and its results supported the theoretical study. The simulation study results also showed that the AUMRTE is work well at the high level of correlation. For the real application, it was applied to the data of the GDP of the Egyptian tourism sector. The results of the application showed that the AUMTE improve the prediction accuracy for the model. In addition, the results of applied were confirmed with the simulation study results.

Declarations

Author contribution statement

Tarek Mahmoud Omara: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data included in article/supp. material/referenced in article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.