On the Performance of Principal Component Liu-Type Estimator under the Mean Square Error Criterion

Abstract

Wu (2013) proposed an estimator, principal component Liu-type estimator, to overcome multicollinearity. This estimator is a general estimator which includes ordinary least squares estimator, principal component regression estimator, ridge estimator, Liu estimator, Liu-type estimator, 𝑟-𝑘 class estimator, and 𝑟-𝑑 class estimator. In this paper, firstly we use a new method to propose the principal component Liu-type estimator; then we study the superior of the new estimator by using the scalar mean squares error criterion. Finally, we give a numerical example to show the theoretical results.

1. Introduction

Consider the multiple linear regression model 𝑦=𝑋𝛽+𝜖, (1) where 𝑦 is an 𝑛×1 vector of observation, 𝑋 is an 𝑛×𝑝 known matrix of rank 𝑝, 𝛽 is a 𝑝×1 vector of unknown parameters, and 𝜖 is an 𝑛×1 vector of disturbances with expectation 𝐸(𝜖)=0 and variance-covariance matrix Cov(𝜖)=𝜎²𝐼_𝑛.

According to the Gauss-Markov theorem, the classical ordinary least squares estimator (OLSE) is obtained as follows: 𝛽=𝑋^′𝑋⁻¹𝑋^′𝑦. (2) The OLSE has been regarded as the best estimator for a long time. However, when multicollinearity is present and the matrix 𝑋^′𝑋 is ill-conditioned, the OLSE is no longer a good estimator. To improve OLSE, many ways have been proposed. One way is to consider biased estimator, such as, principal component regression estimator [1], ridge estimator [2], Liu estimator [3], Liu-type estimator [4], two-parameter ridge estimator [5], 𝑟-𝑘 class estimator [6], 𝑟-𝑑 class estimator [7], and modified 𝑟-𝑘 class estimator [8].

An alternative method to overcome the multicollinearity is to consider the restrictions. Xu and Yang [9] introduced a stochastic restricted Liu estimator; Li and Yang [10] introduced a stochastic restricted ridge estimator.

To overcome multicollinearity, Hoerl and Kennard [2] solve the following problem: min_𝛽𝑦−𝑋𝛽^′𝑦−𝑋𝛽+𝑘𝛽𝛽^′−𝑐, (3) where 𝑘 is a Lagrangian multiplier and 𝑐 is a constant, and obtain the ridge estimator (RE): 𝛽(𝑘)=𝑋^′𝑋+𝑘𝐼⁻¹𝑋^′𝑦,𝑘>0. (4)

Liu [3] introduced the Liu estimator (LE): 𝛽(𝑑)=𝑋^′𝑋+𝐼⁻¹𝑋^′𝑦+𝑑𝛽,0<𝑑<1, (5) where 𝛽 is OLSE. This estimator can be obtained by solving the following problem: min_𝛽𝑦−𝑋𝛽^′𝑦−𝑋𝛽+𝛽−𝑑𝛽𝛽−𝑑𝛽^′. (6) This estimator can also be obtained by the following ways. Suppose that 𝛽 satisfied 𝑑𝛽=𝛽+𝑒. Then, we use the mixed method [11]; we can obtain the Liu estimator.

Recently, Huang et al. [4] introduced a Liu-type estimator which includes the OLSE, RE, and LE, defined as follows: 𝛽(𝑘,𝑑)=𝑋^′𝑋+𝑘𝐼⁻¹𝑋^′𝑦+𝑑𝛽,𝑘>0,0<𝑑<1, (7) where 𝛽 is OLSE. This estimator can be obtained by solving the following problem: min_𝛽𝑦−𝑋𝛽^′+𝑘𝑦−𝑋𝛽_1/2𝑑𝛽𝛽−𝑘^1/2𝑘^1/2𝑑𝛽𝛽−𝑘^1/2′. (8)

Let us consider the following transformation for the model (1): 𝑦=𝑋𝑇𝑇^′𝛽+𝜖=𝑍𝛼+𝜖, (9) where 𝑋^′𝑋=𝑇_𝑟,𝑇_𝑝−𝑟Λ_𝑟00Λ_𝑝−𝑟𝑇^′_𝑟𝑇^′_𝑝−𝑟. (10)Λ_𝑟 and Λ_𝑝−𝑟 are diagonal matrices such that that the main diagonal elements of the 𝑟×𝑟 matrix Λ_𝑟 are the 𝑟 largest eigenvalues of 𝑋^′𝑋, while Λ_𝑝−𝑟 are the remaining 𝑝−𝑟 eigenvalues. The 𝑝×𝑝 matrix 𝑇=(𝑇_𝑟,𝑇_𝑝−𝑟) is orthogonal with 𝑇_𝑟=(𝑡₁,𝑡₂,…,𝑡_𝑟) consisting of its first 𝑟 columns and 𝑇_𝑝−𝑟=(𝑡_𝑟+1,𝑡_𝑟+2,…,𝑡_𝑝) consisting of the remaining 𝑝−𝑟 columns of the matrix 𝑇. The PCRE for 𝛽 can be written as 𝛽_𝑟=𝑇_𝑟𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟⁻¹𝑇^′_𝑟𝑋^′𝑦. (11)

Baye and Parker [6] introduced the application of ridge approach to improve the PCR estimator, namely, 𝑟-𝑘 class estimator as 𝛽_𝑟(𝑘)=𝑇_𝑟𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟+𝑘𝐼_𝑟⁻¹𝑇^′_𝑟𝑋^′𝑦,𝑘>0. (12)

Alternatively, Kaçıranlar and Sakallıoğlu [7] introduced the 𝑟-𝑑 class estimator which is the combination of the LE and the PCRE, which is defined as follows: 𝛽_𝑟(𝑑)=𝑇_𝑟𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟+𝐼_𝑟⁻¹𝑇^′_𝑟𝑋^′𝑦+𝑑𝑇^′_𝑟𝛽_𝑟,0<𝑑<1. (13)

Wu [12] proposed the principal component Liu-type estimator (PCTTE), which is defined as 𝛽_𝑟(𝑘,𝑑)=𝑇_𝑟𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟+𝑘𝐼_𝑟⁻¹𝑇^′_𝑟𝑋^′𝑦+𝑑𝑇^′_𝑟𝛽_𝑟,𝑘>0,0<𝑑<1. (14)

In this paper, firstly we use a new method to propose the principal component Liu-type estimator. Then, we show that, under certain conditions, the PCTTE is superior to the related estimator in the mean square error criterion. Finally, we give a numerical example to illustrate the theoretical results.

2. The Principal Component Liu-Type Estimator

Using the symbols in (9) and (10), (1) can be written as follows: 𝑦=𝑋𝑇_𝑟𝑇^′_𝑟𝛽+𝑋𝑇_𝑝−𝑟𝑇^′_𝑝−𝑟𝛽+𝜖=𝑍_𝑟𝛼_𝑟+𝑍_𝑝−𝑟𝛼_𝑝−𝑟+𝜖. (15) The PCRE can be obtained by omitted 𝑍_𝑝−𝑟𝛼_𝑝−𝑟, and the model (15) reduced to: 𝑦=𝑍_𝑟𝛼_𝑟+𝜖. (16) Then, solve the following problem: 𝑦−𝑍_𝑟𝛼_𝑟^′𝑦−𝑍_𝑟𝛼_𝑟, (17) we obtain 𝛼_𝑟=𝑍^′_𝑟𝑍_𝑟⁻¹𝑍^′_𝑟𝑦. (18) Then, transforming 𝛼_𝑟 to the original parameter space, we can get the PCRE of parameter 𝛽.

Now, we give a method to obtain the principal component Liu-type estimator. Let 𝑐 be a constant and 𝑘 a Lagrangian multiplier, minimizing 𝑦−𝑍_𝑟𝛼_𝑟^′𝑦−𝑍_𝑟𝛼_𝑟+1𝑘𝑘𝛼_𝑟−𝑑𝛼_𝑟𝑘𝛼_𝑟−𝑑𝛼_𝑟^′−𝑐, (19) where 𝛼_𝑟=(𝑍^′_𝑟𝑍_𝑟)⁻¹𝑍^′_𝑟𝑦. Then we get 𝛼_𝑟(𝑘,𝑑)=𝑍^′_𝑟𝑍_𝑟+𝑘𝐼_𝑟⁻¹𝑍^′_𝑟𝑦+𝑑𝛼_𝑟. (20) After transforming back to original parameters pace, we obtain 𝛽_𝑟(𝑘,𝑑)=𝑇_𝑟𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟+𝑘𝐼_𝑟⁻¹×𝑇^′_𝑟𝑋^′𝑦+𝑑𝑇^′_𝑟𝛽_𝑟,𝑘>0,0<𝑑<1. (21) This estimator can also be got by minimizing the function 𝑘𝛼_𝑟−𝑑𝛼_𝑟𝑘𝛼_𝑟−𝑑𝛼_𝑟^′+𝑘𝑦−𝑍_𝑟𝛼_𝑟^′𝑦−𝑍_𝑟𝛼_𝑟−𝑐. (22) It is easy to see that the new estimator 𝛽_𝑟(𝑘,𝑑) has the following properties:(1)𝛽_𝑟(1,1)=𝛽_𝑟=𝑇_𝑟(𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟)⁻¹𝑇^′_𝑟𝑋^′𝑦 is the PCRE;(2)𝛽_𝑝(1,1)=𝛽_OLS=(𝑋^′𝑋)⁻¹𝑋^′𝑦 is the OLSE;(3)𝛽_𝑟(1,𝑑)=𝛽_𝑟(𝑑)=𝛽_𝑟(𝑑)=𝑇_𝑟(𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟+𝐼_𝑟)⁻¹(𝑇^′_𝑟𝑋^′𝑦+𝑑𝑇^′_𝑟𝛽_𝑟) is the 𝑟-𝑑 class estimator;(4)𝛽_𝑝(1,𝑑)=𝛽(𝑑)=(𝑋^′𝑋+𝐼)⁻¹(𝑋^′𝑦+𝑑𝛽_OLS) is the LE;(5)𝛽_𝑟(𝑘,0)=𝛽_𝑟(𝑘)=𝑇_𝑟(𝑇^′_𝑟𝑋^′𝑋𝑇_𝑟+𝑘𝐼_𝑟)⁻¹𝑇^′_𝑟𝑋^′𝑦 is the 𝑟-𝑘 class estimator;(6)𝛽_𝑝(𝑘,0)=𝛽(𝑘)=(𝑋^′𝑋+𝑘𝐼)⁻¹𝑋^′𝑦 is the RE;(7)𝛽_𝑝(𝑘,𝑑)=𝛽(𝑘,𝑑)=(𝑋^′𝑋+𝑘𝐼)⁻¹(𝑋^′𝑦+𝑘𝑑𝛽_OLS) is the LTE.

3. Superiority of the Principal Component Liu-Type Estimator over Some Estimators in the Mean Square Error Criterion

The mean square error (MSE) of an estimator 𝛽 is defined as 𝑀𝛽=𝐸𝛽−𝛽^′𝛽+𝛽𝛽−𝛽=tr𝑉Bias^′Bias𝛽, (23) where 𝑉(𝛽) is the dispersion matrix and Bias(𝛽) is the bias vector. For two given estimators 𝛽₁ and 𝛽₂, the estimator 𝛽₂ is said to be superior to 𝛽₁ in the MSE criterion, if and only if Δ𝛽₁,𝛽₂=𝑀𝛽₁−𝑀𝛽₂≥0. (24)

3.1. 𝛽_𝑟(𝑘,𝑑) versus 𝛽_𝑟

Firstly, we compute that 𝑉𝛽_𝑟(𝑘,𝑑)=𝜎²𝑇_𝑟𝑆_𝑟(𝑘)⁻¹𝑆_𝑟(𝑑)Λ_𝑟⁻¹𝑆_𝑟(𝑑)𝑆_𝑟(𝑘)⁻¹𝑇^′_𝑟,𝐸𝛽_𝑟(𝑘,𝑑)=𝑇_𝑟𝑆_𝑟(𝑘)⁻¹𝑆_𝑟(𝑑)𝑇^′_𝑟𝛽, (25) where 𝑆_𝑟(𝑘)=Λ_𝑟+𝑘𝐼_𝑟. Then, the mean square error (MSE) of 𝛽_𝑟(𝑘,𝑑) is given as follows: 𝑀𝛽_𝑟(𝑘,𝑑)=tr𝑉𝛽_𝑟()+𝛽𝑘,𝑑Bias_𝑟(𝑘,𝑑)^′Bias𝛽_𝑟()=𝑘,𝑑_𝑟𝑖=1𝜎²𝜆_𝑖+𝑑²+(𝑘−𝑑)²𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+𝑘²+^𝑝_𝑖=𝑟+1𝛼²_𝑖. (26)

Let 𝑘=𝑑 in (26); we obtain the MSE of 𝛽_𝑟 as follows: 𝑀𝛽_𝑟=^𝑟_𝑖=1𝜎²𝜆_𝑖+^𝑝_𝑖=𝑟+1𝛼²_𝑖. (27)

Now we consider the following difference: 𝑀𝛽_𝑟(𝑘,𝑑)−𝑀𝛽_𝑟=^𝑟_𝑖=1𝜎²𝜆_𝑖+𝑑²+(𝑘−𝑑)²𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+𝑘²+^𝑝_𝑖=𝑟+1𝛼²_𝑖−^𝑟_𝑖=1𝜎²𝜆_𝑖+^𝑝_𝑖=𝑟+1𝛼²_𝑖=^𝑟_𝑖=1()𝛼𝑑−𝑘^2𝑖𝜆_𝑖+𝜎²𝑑+2𝜎²𝜆_𝑖+𝜎²−𝛼²_𝑖𝜆_𝑖𝑘𝜆_𝑖𝜆_𝑖+𝑘². (28) If 𝜎²−𝛼²_𝑖𝜆_𝑖>0, then when 0<𝑑<𝑘,  𝑀(𝛽_𝑟(𝑘,𝑑))−𝑀(𝛽_𝑟)<0. If 𝜎²−𝛼²_𝑖𝜆_𝑖<0, then when 0<𝑑<𝑘<min{((𝛼²_𝑖𝜆_𝑖+𝜎²)𝑑+2𝜎²𝜆_𝑖)/(𝛼²_𝑖𝜆_𝑖−𝜎²)} and max{((𝛼²_𝑖𝜆_𝑖+𝜎²)𝑑+2𝜎²𝜆_𝑖)/(𝛼²_𝑖𝜆_𝑖−𝜎²)}<𝑘<𝑑<1,  𝑀(𝛽_𝑟(𝑘,𝑑))−𝑀(𝛽_𝑟)<0. So we have the following theorem.

Theorem 1. The estimator 𝛽_𝑟(𝑘,𝑑) is superior to the estimator for 𝛽_𝑟 under the mean square error criterion for:(a)0<𝑑<𝑘 if 𝜎²−𝛼²_𝑖𝜆_𝑖>0 for all 𝑖=1,…,𝑟,(b)0<𝑑<𝑘<min{((𝛼²_𝑖𝜆_𝑖+𝜎²)𝑑+2𝜎²𝜆_𝑖)/(𝛼²_𝑖𝜆_𝑖−𝜎²)} and max{((𝛼²_𝑖𝜆_𝑖+𝜎²)𝑑+2𝜎²𝜆_𝑖)/(𝛼²_𝑖𝜆_𝑖−𝜎²)}<𝑘<𝑑<1, 𝑖=1,…,𝑟 if 𝜎²−𝛼²_𝑖𝜆_𝑖<0 for all 𝑖=1,…,𝑟.

3.2. 𝛽_𝑟(𝑘,𝑑) versus 𝛽_𝑟(𝑘)

From the definition of the 𝛽_𝑟(𝑘), we know that let 𝑑=0 in 𝛽_𝑟(𝑘,𝑑), and we obtain the 𝛽_𝑟(𝑘).

Theorem 2. Let 𝑘𝛼²_𝑖−𝜎²>0 for all 𝑖=1,…,𝑟. Then, there exists a strictly positive 𝑑 such that 𝛽_𝑟(𝑘,𝑑) is superior to 𝛽_𝑟(𝑘) in the mean square error criterion for 0<𝑑<(∑^𝑟_𝑖=1((𝑘𝛼²_𝑖−𝜎²)/(𝜆_𝑖+𝑘)²))/(∑^𝑟_𝑖=1((𝜎²+𝛼²_𝑖𝜆_𝑖)/(𝜆_𝑖(𝜆_𝑖+𝑘)²)))<1.

Proof. We know that 𝑀(𝛽_𝑟(𝑘))=𝑀𝛽_𝑟(𝑘,0), so that by continuity it is sufficient to show that 𝑀(𝛽_𝑟(𝑘,𝑑)) decreasing in the neighborhood of 𝑑=0.
Performing the calculus for fixed 𝑘, we can see that 𝜕𝑀𝛽_𝑟(𝑘,𝑑)𝜕𝑑=2^𝑟_𝑖=1𝜎²𝜆_𝑖+𝑑−(𝑘−𝑑)𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+𝑘². (29) So when 0<𝑑<(∑^𝑟_𝑖=1((𝑘𝛼²_𝑖−𝜎²)/(𝜆_𝑖+𝑘)²))/(∑^𝑟_𝑖=1((𝜎²+𝛼²_𝑖𝜆_𝑖)/(𝜆_𝑖(𝜆_𝑖+𝑘)²)))<1, (𝜕𝑀(𝛽_𝑟(𝑘,𝑑)))/(𝜕𝑑)<0; that is to say, 𝑀(𝛽_𝑟(𝑘,𝑑))<𝑀(𝛽_𝑟(𝑘,0))=𝑀(𝛽_𝑟(𝑘)). The proof of Theorem 2 is completed.

3.3. 𝛽_𝑟(𝑘,𝑑) versus 𝛽_𝑟(𝑑)

From the definition of the 𝛽_𝑟(𝑑), we know that let 𝑘=1 in 𝛽_𝑟(𝑘,𝑑), and we obtain that 𝛽_𝑟(𝑑), 𝑀𝛽_𝑟(𝑑)=^𝑟_𝑖=1𝜎²𝜆_𝑖+𝑑²+(1−𝑑)²𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²+^𝑝_𝑖=𝑟+1𝛼²_𝑖. (30)

Now, we discuss the following difference: Δ=𝑀𝛽_𝑟(𝑘,𝑑)−𝑀𝛽_𝑟(𝑘)=^𝑟_𝑖=1𝜎²𝜆_𝑖+𝑑²+(𝑘−𝑑)²𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+𝑘²+^𝑝_𝑖=𝑟+1𝛼²_𝑖−^𝑟_𝑖=1𝜎²𝜆_𝑖+𝑑²+(1−𝑑)²𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²+^𝑝_𝑖=𝑟+1𝛼²_𝑖=^𝑟_𝑖=1𝜎²(1−𝑘)𝜆_𝑖+𝑑𝜆_𝑖+𝑑2𝜆_𝑖+𝑘+1𝜆_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²−^𝑟_𝑖=1𝜎²(1−𝑘)𝜆_𝑖+𝑑𝛼²_𝑖𝜆_𝑖×(𝑘+1−2𝑑)𝜆_𝑖×𝜆+2𝑘−𝑑−𝑑𝑘_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²⁻¹. (31) Then by (31), if we want Δ<0, then 𝑑<𝑑₁=^𝑟_𝑖=12𝜆_𝑖+𝑘+1𝜎²+2𝛼²_𝑖𝜆²_𝑖+(1+𝑘)𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²×^𝑟_𝑖=1𝜆_𝑖𝜎²(2𝜆_𝑖+𝑘+1)−𝛼²_𝑖𝜆²_𝑖(𝑘+1)−2𝑘𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²⁻¹ (32) when 𝑘<1, it is easy to know that 𝑑₁ is always less than 1, and 𝑑₁ is to be bigger than 0, if ^𝑟_𝑖=1𝜆_𝑖𝜎²2𝜆_𝑖+𝑘+1−𝛼²_𝑖𝜆²_𝑖(𝑘+1)−2𝑘𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²>0. (33) Then, we get 𝑘>2𝜆_𝑖+1𝜎²−𝛼²_𝑖𝜆_𝑖𝛼²_𝑖𝜆_𝑖+2−𝜎² (34) for 𝛼²_𝑖(𝜆_𝑖+2)−𝜎²>0 with all 𝑖=1,…,𝑟.

Thus, we obtain the following theorem.

Theorem 3. If 𝛼²_𝑖(𝜆_𝑖+2)−𝜎²>0 and 𝑘>{max(((2𝜆_𝑖+1)𝜎²−𝛼²_𝑖𝜆_𝑖)/(𝛼²_𝑖(𝜆_𝑖+2)−𝜎²)),0} for all 𝑖=1,…,𝑟, then the 𝛽_𝑟(𝑘,𝑑) is better than the 𝛽_𝑟(𝑑) in the mean square error sense for 𝑘<1 and 𝑑<^𝑟_𝑖=12𝜆_𝑖+𝑘+1𝜎²+2𝛼²_𝑖𝜆²_𝑖+(1+𝑘)𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²×^𝑟_𝑖=1𝜆_𝑖𝜎²2𝜆_𝑖+𝑘+1−𝛼²_𝑖𝜆²_𝑖(𝑘+1)−2𝑘𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+1²𝜆_𝑖+𝑘²⁻¹. (35)

3.4. 𝛽_𝑟(𝑘,𝑑) versus 𝛽

In this subsection, we will give the comparison of the 𝛽_𝑟(𝑘,𝑑) and 𝛽 under the mean square error criterion.

Let 𝑘=𝑑 and 𝑟=𝑝 in (26); we obtain the MSE of 𝛽_𝑟 as follows: 𝑀𝛽=^𝑝_𝑖=1𝜎²𝜆_𝑖. (36)

In order to compare the 𝛽_𝑟(𝑘,𝑑) and 𝛽, now we consider the following difference: 𝑀𝛽_𝑟()𝛽=𝑘,𝑑−𝑀_𝑟𝑖=1𝜎²𝜆_𝑖+𝑑²+(𝑘−𝑑)²𝛼²_𝑖𝜆_𝑖𝜆_𝑖𝜆_𝑖+𝑘²+^𝑝_𝑖=𝑟+1𝛼²_𝑖−^𝑝_𝑖=1𝜎²𝜆_𝑖=^𝑟_𝑖=1()×𝛼𝑑−𝑘_2𝑖𝜆_𝑖+𝜎²𝑑+2𝜎²𝜆_𝑖+𝜎²−𝛼²_𝑖𝜆_𝑖𝑘𝜆_𝑖𝜆_𝑖+𝑘²+^𝑝_𝑖=𝑟+1𝛼²_𝑖𝜆_𝑖−𝜎²𝜆_𝑖 (37) when 𝑘>𝑑>𝑑₂=(∑^𝑟_𝑟=1(((𝛼²_𝑖𝜆_𝑖−𝜎²)𝑘−2𝜎²𝜆_𝑖)/(𝜆_𝑖(𝜆_𝑖+𝑘)²)))/(∑^𝑟_𝑟=1((𝛼²_𝑖𝜆_𝑖+𝜎²)/(𝜆_𝑖(𝜆_𝑖+𝑘)²))), then ^𝑟_𝑖=1()𝛼𝑑−𝑘^2𝑖𝜆_𝑖+𝜎²𝑑+2𝜎²𝜆_𝑖+𝜎²−𝛼²_𝑖𝜆_𝑖𝑘𝜆_𝑖𝜆_𝑖+𝑘²<0. (38) Since the lower bound of 𝑑 is less than 1, the lower bound of 𝑑 may be less than 0. If the lower bound of of 𝑑 is less than 0, then we can choose any 𝑑 in [0,1]. Thus, we can get the following theorem.

Theorem 4. (1)If 𝛼²_𝑖𝜆_𝑖−𝜎²<0  for all 𝑖=1,2,…,𝑝, then 𝛽_𝑟(𝑘,𝑑) is superior to 𝛽 in the mean square error sense for any 𝑘>0 and 0<𝑑<1.(2)If 𝛼²_𝑖𝜆_𝑖−𝜎²<0 for some 𝑖=1,2,…,𝑟, then 𝛽_𝑟(𝑘,𝑑) is superior to 𝛽 in the mean square error sense for 𝑘>𝑑>(∑^𝑟_𝑟=1(((𝛼²_𝑖𝜆_𝑖−𝜎²)𝑘−2𝜎²𝜆_𝑖)/(𝜆_𝑖(𝜆_𝑖+𝑘)²)))/(∑^𝑟_𝑟=1((𝛼²_𝑖𝜆_𝑖+𝜎²)/(𝜆_𝑖(𝜆_𝑖+𝑘)²))) when ∑^𝑝_𝑖=𝑟+1((𝛼²_𝑖𝜆_𝑖−𝜎²)/𝜆_𝑖)<0.

3.5. 𝛽_𝑟(𝑘,𝑑) versus 𝛽(𝑘,𝑑)

In this subsection, we will compare 𝛽_𝑟(𝑘,𝑑) with 𝛽(𝑘,𝑑) in the mean square error sense.

Define 𝜐_𝑖=(𝜆_𝑖+𝑑)/(𝜆_𝑖+𝑘), then the mean square error of 𝛽_𝑟(𝑘,𝑑) can be written as 𝑀𝛽_𝑟(𝑘,𝑑)=^𝑟₁₌₁𝜎²𝜐²_𝑖𝜆_𝑖+𝜐_𝑖−1²𝛼²_𝑖+^𝑝_𝑖=𝑟+1𝛼²_𝑖. (39) Firstly, we discuss the difference between 𝑀(𝛽_𝑟(𝑘,𝑑)) and 𝑀(𝛽_𝑟−1(𝑘,𝑑)): 𝑀𝛽_𝑟(𝑘,𝑑)−𝑀𝛽_𝑟−1(𝑘,𝑑)=𝜐_𝑟𝜐_𝑟𝜎²𝜆_𝑟+𝛼²_𝑟−2𝛼²_𝑟=𝜐_𝑟𝜆_𝑖𝑑𝜎+𝑘𝜆²_𝑟+𝛼²_𝑟+𝜎²−𝛼²_𝑟𝜆_𝑟−2𝑘𝛼²_𝑟. (40) So when 𝜎²<𝛼²_𝑟𝜆_𝑟+2𝑘𝛼²_𝑟, if 1>𝑑>((𝛼²_𝑟𝜆_𝑟+2𝑘𝛼²_𝑟−𝜎²)/((𝜎²/𝜆_𝑟)+𝛼²_𝑟))>0, then 𝑀(𝛽_𝑟(𝑘,𝑑))−𝑀(𝛽_𝑟−1(𝑘,𝑑))>0.

If 0<𝑑<((𝛼²_𝑟𝜆_𝑟+2𝑘𝛼²_𝑟−𝜎²)/((𝜎²/𝜆_𝑟)+𝛼²_𝑟))<1, then 𝑀(𝛽_𝑟(𝑘,𝑑))−𝑀(𝛽_𝑟−1(𝑘,𝑑))<0.

Then we have the following theorem.

Theorem 5. (1)If 𝜎²<𝛼²_𝑖𝜆_𝑖+2𝑘𝛼²_𝑖 for some 𝑖∈𝑁_𝑝, then there exists a nonnegative 𝑑₃ such that 𝑀(𝛽(𝑘,𝑑))>𝑀(𝛽_𝑟(𝑘,𝑑)) for 0<𝑑₃<𝑑<1, where 𝑁_𝑝=𝑟+1,…,𝑝.(2)If 𝜎²<𝛼²_𝑖𝜆_𝑖+2𝑘𝛼²_𝑖 for all 𝑖∈𝑁_𝑝, then there exists a nonnegative 𝑑₃ such that 𝑀(𝛽(𝑘,𝑑))<𝑀(𝛽_𝑟(𝑘,𝑑)) for 0<𝑑<𝑑₄<1, where 𝑁_𝑝=𝑟+1,…,𝑝, where 𝑑₃=min_{𝑖∈𝑁𝑝}1,max_{𝑖∈𝑁𝑝}𝛼²_𝑖𝜆_𝑖+2𝑘𝛼²_𝑖−𝜎²𝜎²/𝜆_𝑖+𝛼²_𝑖,𝑑₄=min_{𝑖∈𝑁𝑝}𝛼²_𝑖𝜆_𝑖+2𝑘𝛼²_𝑖−𝜎²𝜎²/𝜆_𝑖+𝛼²_𝑖. (41)

4. Numerical Example

To illustrate our theoretical results, firstly we use a numerical example to investigate the estimators studied in the dataset originally due to Gruber [13], and later considered by Akdeniz and Erol [14]. We assemble the data as follows: ,.𝑋=1.92.21.93.71.82.22.03.81.82.42.13.61.82.42.23.82.02.52.33.82.12.62.43.72.12.62.63.82.22.62.64.02.32.82.83.72.32.72.83.8𝑦=2.32.22.22.32.42.52.62.62.72.7 (42) Firstly, we can compute the OLSE of 𝛽 as follows: 𝛽=(0.6921,0.6258,−0.1154,0.2866,0.0256)^′. (43) For the OLSE, PCRE, 𝑟-𝑘 class estimator, 𝑟-𝑑 class estimator, Liu-type estimator (LTE), and new estimator (PCTTE), their estimated mean square error (MSE) values are obtained by replacing all unknown model parameters by their, respectively, least squares estimators in corresponding expressions.

Firstly, we see the comparison of the OLSE and the PCTTE. When 𝑑=0.6 is fixed, then if the values of 𝑘 is big then the new estimator has smaller MSE values than OLSE; that is to say, the new estimator is better than the OLSE. So we see that the new estimator improved the OLSE.

From Figure 3, we see that when 𝑑, fixed, if 0<𝑑<𝑘, then the new estimator is better than the PCRE. In Theorem 1, we see that if 0<𝑑<𝑘, the new estimator is better. For the numerical example, when 0<𝑑<𝑘, the new estimator is more efficient than the PCRE.

By Figures 1, 2, 3, 4, 5, 6, 7, and 8, we see that when 0<𝑑<𝑘, then new estimator is better than the LTE. So in practice, we can choose bigger 𝑘 and smaller 𝑑.

Figure 1 

The MSE values of OLSE and PCTTE when 𝑑=0.6.

Figure 2 

The MSE values of OLSE and PCTTE when 𝑘=0.56.

Figure 3 

The MSE values of PCRE and PCTTE when 𝑑=0.6.

Figure 4 

The MSE values of PCRE and PCTTE when 𝑘=0.56.

Figure 5 

The MSE values of 𝑟-𝑘 class estimator and PCTTE when 𝑑=0.1.

Figure 6 

The MSE values of 𝑟-𝑑 class estimator and PCTTE when 𝑘=0.9.

Figure 7 

The MSE values of LTE and PCTTE when 𝑘=0.8.

Figure 8 

The MSE values of LTE and PCTTE when 𝑑=0.3.

5. Conclusion

In this paper, we use a new method to propose the principal component Liu-type estimator. Then, we discuss the superiority of the new estimator with the OLSE, PCRE, 𝑟-𝑘 class estimator, 𝑟-𝑑 class estimator, and Liu-type estimator in the sense of mean square error. We also give a method to choose the biasing parameters. Finally, we give a numerical example to illustrate the performance the various estimators.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to three anonymous referees for their valuable comments which improved the quality of the paper. This work was supported by the Scientific Research Foundation of Chongqing University of Arts and Sciences (Grants nos. Y2013SC43, R2013SC12), Program for Innovation Team Building at Institutions of Higher Education in Chongqing (Grant no. KJTD201321), and the National Natural Science Foundation of China (nos. 11201505, 71271227).