An Unbiased Two-Parameter Estimation with Prior Information in Linear Regression Model

Abstract

We introduce an unbiased two-parameter estimator based on prior information and two-parameter estimator proposed by Özkale and Kaçıranlar, 2007. Then we discuss its properties and our results show that the new estimator is better than the two-parameter estimator, the ordinary least squares estimator, and explain the almost unbiased two-parameter estimator which is proposed by Wu and Yang, 2013. Finally, we give a simulation study to show the theoretical results.

1. Introduction

Consider the following linear regression model: 𝑌=𝑋𝛽+𝜖, (1) where 𝑌 shows an 𝑛×1 vector of observations on the dependent variable, 𝑋 shows an 𝑛×𝑝 known design matrix of rank 𝑝, 𝛽 shows a 𝑝×1 vector of unknown regression coefficients, and 𝜖 shows an 𝑛×1 vector of disturbances with 𝐸(𝜖)=0 and variance-covariance matrix Cov(𝜖)=𝜎²𝐼_𝑛.

As we all know, the ordinary least squares (OLS) estimator 𝛽_OLS=(𝑋^′𝑋)⁻¹𝑋^′𝑌 has been regarded as the best estimator for a long time. However, when the multicollinearity occurs, the OLS estimator is no longer a good estimator. To treat this problem, many approaches have been presented. One method is to consider the biased estimator, such as Hoerl and Kennard [1], Swindel [2], Farebrother [3], Liu [4], Sakallog lu and Akdeniz [5], Özkale and Kaçıranlar [6, 7], Yang and Chang [8], and Wu and Yang [9, 10]. Although these biased estimators can treat multicollinearity, these estimators have big bias. In order to reduce the bias, Crouse et al. [11] and Sakallog lu and Akdeniz [5] based on ridge estimator and Liu estimator proposed the unbiased ridge estimator and unbiased Liu estimator with prior information, respectively. The unbiased ridge estimator and unbiased Liu estimator not only can deal with multicollinearity, but also have no bias.

In this paper, we will introduce an unbiased two-parameter estimator with prior information and show some properties of the new estimator.

The reminder of this paper is organized as follows. In Section 2, we give the unbiased two-parameter estimator and comparisons with OLS, two-parameter estimator proposed by Özkale and Kaçıranlar [7], and almost unbiased two-parameter estimator proposed by Wu and Yang [9] in the sense of MMSE criterion. The estimators of the parameters 𝑘 and 𝑑 are proposed in Section 3. A simulation study is given to explain the theoretical results in Section 4 and some conclusion remarks are given in Section 5.

2. Analysis of Unbiased Two-Parameter Estimator with Prior Information

In this section, we also consider the general linear regression model (1) and thus the 𝛽_OLS∼𝑁(𝛽,𝜎²𝑆⁻¹) for 𝑆=𝑋^′𝑋.

Crouse et al. [11] presented the unbiased ridge estimator based on the ridge estimator and prior information 𝐽, which is defined as follows: 𝛽(𝑘𝐼,𝐽)=𝑋^′𝑋+𝑘𝐼⁻¹𝑋^′𝑌+𝑘𝐽 (2) with 𝐽 being uncorrelated with 𝛽_OLS and 𝐽∼𝑁(𝛽,𝑉). In (2), 𝑉=(𝜎²/𝑘)𝐼. And in (2) the prior information 𝐽 is a random vector for specified mean and covariance.

The two-parameter estimator proposed by Özkale and Kaçıranlar [7] is defined as follows: 𝛽(𝑘,𝑑)=𝑋^′𝑋+𝑘𝐼⁻¹𝑋^′𝑌+𝑘𝑑𝛽_OLS=𝐹_𝑘𝑑𝛽_OLS, (3) where 𝛽_OLS is the OLS estimator, 𝐹_𝑘𝑑=(𝑋^′𝑋+𝑘𝐼)⁻¹(𝑋^′𝑋+𝑘𝑑𝐼), and 𝑘>0, 0<𝑑<1.

Based on the two-parameter estimator, Wu and Yang [9] proposed an almost unbiased two-parameter estimator: 𝛽_AUTP(𝑘,𝑑)=𝐼−𝑘²(1−𝑑)²𝑋^′𝑋+𝑘𝐼⁻²𝛽_OLS. (4) Now we study the following convex estimator: 𝛽(𝐶,𝐽)=𝐶𝛽_OLS+(𝐼−𝐶)𝐽, (5) with 𝐶 presenting a 𝑝×𝑝 matrix and 𝐼 showing the 𝑝×𝑝 identity matrix. Then we can compute the mean squared error (MSE) of 𝛽(𝐶,𝐽): MSE𝛽(𝐶,𝐽)=𝜎²𝐶𝑆⁻¹𝐶^′+(𝐼−𝐶)𝑉(𝐼−𝐶)^′. (6) Now we find a matrix such that MSE{𝛽(𝐶,𝐽)} reaches a minimum. Solve 𝜕MSE𝛽(𝐶,𝐽)𝜎𝜕𝐶=2𝐶²𝑆⁻¹+𝑉−2𝑉=0. (7) Then we obtain 𝐶=𝑉(𝜎²𝑆⁻¹+𝑉)⁻¹. Accordingly, we get 𝑉=𝜎²(𝐼−𝐶)⁻¹𝐶𝑆⁻¹.

Now we can define the following estimator: 𝛽𝐹_𝑘𝑑,𝐽=𝐹_𝑘𝑑𝛽_OLS+𝐼−𝐹_𝑘𝑑𝐽=𝛽(𝑘,𝑑)+𝐼−𝐹_𝑘𝑑𝐽. (8) Hence, for optimal value of 𝐶 under the minimum MSE, the optimal convex estimator 𝛽(𝐶,𝐽) is an unbiased estimator of 𝛽.

For (8), since 𝐹_𝑘𝑑=(𝑋^′𝑋+𝑘𝐼)⁻¹(𝑋^′𝑋+𝑘𝑑𝐼), then we get 𝑉=(𝜎²/𝑘(1−𝑑))(𝑆+𝑘𝑑𝐼)𝑆⁻¹. Then 𝐽∼𝑁(𝛽,(𝜎²/𝑘(1−𝑑))(𝑆+𝑘𝑑𝐼)𝑆⁻¹) for 𝑘>0, 0<𝑑<1.

For (8), it is easy to see that 𝛽(𝐹_𝑘𝑑,𝐽) is an unbiased estimator of 𝛽 and we call this estimator as UTP estimator.

Then in the following section we will give the comparisons of the new estimator with the OLS estimator, the TP estimator, and the AUTP estimator in the matrix mean squared error. Firstly, we give the definition of the matrix mean squared error (MMSE).

The matrix mean squared error (MMSE) is denoted as follows: MMSE(𝑏)=𝐸𝑏−𝛽𝑏−𝛽^′=𝐷(𝑏)+[bias(𝑏),bias(𝑏)]^′, (9) where 𝑏 shows an estimator of 𝛽 and 𝐷(𝑏) and bias(𝑏) present the dispersion matrix and bias vector of 𝑏, respectively.

The mean squared error is denoted as MSE(𝑏)=tr{MMSE(𝑏)}.

Lemma 1. Let 𝑏₁ and 𝑏₂ be two estimators of 𝛽. Then 𝑏₂ is called MMSE superior to 𝑏₁ if 𝑀𝑀𝑆𝐸𝑏₁−𝑀𝑀𝑆𝐸𝑏₂≥0. (10)

Lemma 2 (see [12]). Let 𝑀 be a positive definite matrix, namely, 𝑀>0, and let 𝛼 be some vector; then 𝑀−𝛼𝛼^′≥0 if and only if 𝛼^′𝑀⁻¹𝛼≤1.

Lemma 3 (see [13]). Suppose that 𝑀 is a positive definite matrix and 𝑁 is an nonnegative definite matrix; then 𝑀−𝑁≥0⟺𝜆_max𝑁𝑀⁻¹≤1. (11)

2.1. Comparison of the OLS Estimator and the Unbiased Two-Parameter (UTP) Estimator

Now we compare the unbiased two-parameter (UTP) estimator with the OLS estimator in the matrix mean squared error (MMSE) sense.

Theorem 4. The unbiased two-parameter estimator always dominates the OLS estimator in the MMSE sense for 𝑘>0 and 0<𝑑<1.

Proof. Since 𝐷𝛽_OLS=𝜎²𝑆⁻¹,bias𝛽_OLS𝐷𝛽𝐹=0,_𝑘𝑑,𝐽=𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹,bias𝐹_𝑘𝑑,𝐽=0, (12) so from the definition of MMSE, we have MMSE𝛽_OLS=𝜎²𝑆⁻¹,MMSE𝛽𝐹_𝑘𝑑,𝐽=𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹. (13)(14) Then from (13) and (14), we obtain that MMSE𝛽_OLS−MMSE𝛽𝐹_𝑘𝑑,𝐽=𝜎²𝑆⁻¹−𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹=𝜎²𝑘(1−𝑑)𝑆⁻¹(𝑆+𝑘𝐼)⁻¹, (15) is a nonnegative definite matrix for 𝑘>0 and 0<𝑑<1.
The proof of Theorem 4 is completed.

2.2. Comparison of TP Estimator and the Unbiased Two-Parameter (UTP) Estimator

Now we state the following theorem to compare the unbiased two-parameter estimator (UTP) with the TP estimator in the sense of MMSE.

Theorem 5. The unbiased two-parameter estimator (UTP) is superior to the TP estimator in the sense of MMSE if and only if 𝛽^′()𝜎𝑆+𝑘𝑑𝐼𝑆𝛽>²𝑘().1−𝑑 (16)

Proof. From the definition of the MMSE, we have 𝛽(𝛽(+𝛽()𝛽()MMSE𝑘,𝑑)=𝐷𝑘,𝑑)bias𝑘,𝑑,bias𝑘,𝑑^′=𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹(𝑆+𝑘𝑑𝐼)(𝑆+𝑘𝐼)⁻¹+𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻¹𝛽𝛽^′(𝑆+𝑘𝐼)⁻¹. (17) Thus, from (14) and (17), we obtain MMSE𝛽𝐹_𝑘𝑑,𝐽−MMSE𝛽(𝑘,𝑑)=𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹(𝑆+𝑘𝑑𝐼)(𝑆+𝑘𝐼)⁻¹+𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻¹𝛽𝛽^′(𝑆+𝑘𝐼)⁻¹=𝑘(1−𝑑)(𝑆+𝑘𝐼)⁻¹𝜎²(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝑘(1−𝑑)𝛽𝛽^′×(𝑆+𝑘𝐼)⁻¹. (18) Since 𝑘>0, 0<𝑑<1 and using Lemma 2, we obtain that MMSE𝛽𝐹_𝑘𝑑,𝐽−MMSE𝛽(𝑘,𝑑) (19) is nonnegative definite matrix if and only if 𝛽^′()𝜎𝑆+𝑘𝑑𝐼𝑆𝛽≤²𝑘().1−𝑑 (20) So we can conclude that the unbiased two-parameter estimator (UTP) is superior to the TP estimator in the sense of MMSE if and only if 𝛽^′()𝜎𝑆+𝑘𝑑𝐼𝑆𝛽>²𝑘().1−𝑑 (21)

2.3. Comparison of AUTP Estimator and the Unbiased Two-Parameter (UTP) Estimator

Now we state the following theorem to compare the unbiased two-parameter estimator (UTP) with the AUTP estimator proposed by Wu and Yang [9] in the sense of MMSE.

Theorem 6. If 𝜆_max([𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²][𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²](𝑆+𝑘𝐼)(𝑆+𝑘𝑑𝐼)⁻¹)≤1, the unbiased two-parameter estimator (UTP) is superior to the AUTP estimator in the sense of MMSE if and only if 𝑏^′₁(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝜎²𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²×𝑆⁻¹𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²≥𝜎². (22)

Proof. By (4), we have 𝐷𝛽_AUTP(𝑘,𝑑)=𝜎²𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²×𝑆⁻¹𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²,bias𝛽_AUTP(𝑘,𝑑)=−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²𝛽. (23) Thus, MMSE𝛽_AUTP(𝑘,𝑑)=𝜎²𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²𝑆⁻¹×𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²+𝑘⁴(1−𝑑)⁴(𝑆+𝑘𝐼)⁻²𝛽𝛽^′(𝑆+𝑘𝐼)⁻². (24) Now we consider the following difference: MMSE𝛽𝐹_𝑘𝑑,𝐽−MMSE𝛽_AUTP(𝑘,𝑑)=𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝜎²𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²×𝑆⁻¹𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²−𝑘⁴(1−𝑑)⁴(𝑆+𝑘𝐼)⁻²𝛽𝛽^′(𝑆+𝑘𝐼)⁻²=𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝜎²𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²×𝑆⁻¹𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²−𝑏₁. (25) Since (𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹>0 and [𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²]𝑆⁻¹[𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²]>0, thus by Lemma 3, if 𝜆_max([𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²][𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²](𝑆+𝑘𝐼)(𝑆+𝑘𝑑𝐼)⁻¹)≤1, then 𝜎²(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝜎²[𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²]𝑆⁻¹[𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²]>0. Then by Lemma 2, if 𝑏^′₁{(𝑆+𝑘𝐼)⁻¹(𝑆+𝑘𝑑𝐼)𝑆⁻¹−𝜎²[𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²]𝑆⁻¹[𝐼−𝑘²(1−𝑑)²(𝑆+𝑘𝐼)⁻²]}≥𝜎², the UTP is better than the AUTP estimator.

3. Estimation of the Parameter 𝑘 and Parameter 𝑑

In this section, we discuss how to estimate the biasing parameters 𝑘 and 𝑑.

3.1. The Estimating of the Biasing Parameter 𝑑

In the definition of the new estimator, the OLS 𝛽_OLS is independent of 𝐽. Then 𝛽_OLS−𝐽∼𝑁(0,𝜎²𝑆⁻¹(𝑆+𝑘𝐼)/𝑘(1−𝑑)) and 𝐸𝛽_OLS−𝐽𝛽_OLS−𝐽^′=𝜎²𝑘()𝑆1−𝑑𝑝+𝑘tr⁻¹. (26) From (26), if 𝜎² is known, for a fixed 𝑘, we can get an unbiased estimator of 𝑑 found as follows: 𝜎𝑑=1−²𝑝+𝑘tr𝑆⁻¹𝑘𝛽_OLS−𝐽𝛽_OLS−𝐽^′. (27) When 𝜎² is unknown, we use the following 𝑠² to estimate 𝜎²: 𝑠²=𝑌−𝑋𝛽_OLS^′𝑌−𝑋𝛽_OLS,𝑛−𝑝 (28) and then an estimate of 𝑑 is 𝑠𝑑=1−²𝑝+𝑘tr𝑆⁻¹𝑘𝛽_OLS−𝐽𝛽_OLS−𝐽^′, (29) where tr(𝑆⁻¹)=∑^𝑝_𝑖=11/𝜆_𝑖 and 𝜆_𝑖 is the eigenvalue of 𝑆.

Note that in (27) and (29) the estimator of 𝑑 may be negative. So when being in this situation, one might try to denote 𝑑=1. Summing up these results, the 𝑑 may be presented as follows.

Case I. Assuming 𝜎² is known,(i)if 𝑘(𝛽_OLS−𝐽)(𝛽_OLS−𝐽)^′−𝜎²[𝑝+𝑘tr(𝑆⁻¹)]>0, then 𝑑^∗𝜎=1−²𝑝+𝑘tr𝑆⁻¹𝑘𝛽_OLS−𝐽𝛽_OLS−𝐽^′; (30)(ii)otherwise 𝑑^∗=1. (31)

Case II. Assuming 𝜎² is unknown,(i)if 𝑘(𝛽_OLS−𝐽)(𝛽_OLS−𝐽)^′−𝑠²[𝑝+𝑘tr(𝑆⁻¹)]>0, then 𝑑^∗𝑠=1−²𝑝+𝑘tr𝑆⁻¹𝑘𝛽_OLS−𝐽𝛽_OLS−𝐽^′; (32)(ii)otherwise 𝑑^∗=1, (33) where 𝑠²=(𝑌−𝑋𝛽_OLS)^′(𝑌−𝑋𝛽_OLS)/(𝑛−𝑝) is an unbiased estimator of 𝜎².

3.2. The Estimating of the Biasing Parameter 𝑘

From (26), if 𝜎² is known, for a fixed 𝑑, an unbiased estimate of 𝑘 is defined as follows: 𝑘=𝑝𝜎²(1−𝑑)𝛽_OLS−𝐽𝛽_OLS−𝐽^′−𝜎²tr𝑆⁻¹. (34) When 𝜎² is unknown, similarly an estimate of 𝑘 is 𝑘=𝑝𝑠²(1−𝑑)𝛽_OLS−𝐽𝛽_OLS−𝐽^′−𝑠²tr𝑆⁻¹. (35) Note that in (34) and (35) the estimator of 𝑘 may be negative. So when being in this situation, one might try to denote 𝑘=0. However, there always exists a 𝑘 such that the unbiased two-parameter estimator 𝛽(𝐹_𝑘𝑑,𝐽) has smaller MSE than 𝛽_OLS. Thus, define 𝑘=𝑝𝑠²/(1−𝑑)(𝛽_OLS−𝐽)(𝛽_OLS−𝐽)^′. With the above discussion, 𝑘 may be presented as follows.

Case I. Assuming 𝜎² is known,(i)if (1−𝑑)(𝛽_OLS−𝐽)(𝛽_OLS−𝐽)^′−𝜎²tr(𝑆⁻¹)>0, then 𝑘^∗=𝑝𝜎²(1−𝑑)𝛽_OLS−𝐽𝛽_OLS−𝐽^′−𝜎²tr𝑆⁻¹; (36)(ii)otherwise 𝑘^∗=𝑝𝜎²(1−𝑑)𝛽_OLS−𝐽𝛽_OLS−𝐽^′. (37)

Case II. Assuming 𝜎² is unknown,(i)if (1−𝑑)(𝛽_OLS−𝐽)(𝛽_OLS−𝐽)^′−𝑠²tr(𝑆⁻¹)>0, then 𝑘^∗=𝑝𝑠²(1−𝑑)𝛽_OLS−𝐽𝛽_OLS−𝐽^′−𝑠²tr𝑆⁻¹; (38)(ii)otherwise 𝑘^∗=𝑝𝑠²(1−𝑑)𝛽_OLS−𝐽𝛽_OLS−𝐽^′, (39)

where 𝑠²=(𝑌−𝑋𝛽_OLS)^′(𝑌−𝑋𝛽_OLS)/(𝑛−𝑝) is an unbiased estimator of 𝜎². In applications there may be other estimates of 𝜎² that may also be used.

It is worthwhile to point that the proposed 𝑘 and 𝑑 provide an unbiased two-parameter estimator of 𝛽 while the two-parameter estimator is biased.

4. A Simulation Study

In this section, we will give a simulation study to explain the theoretical results. Following McDonald and Galarneau [14], the explanatory variables are produced using the following device: 𝑥_𝑖𝑗=1−𝑟²𝑧_𝑖𝑗+𝑟𝑧_𝑖(𝑝+1),𝑖=1,…,𝑛,𝑗=1,…,𝑝, (40) where 𝑧_𝑖𝑗 and 𝑧_𝑖(𝑝+1) show independent standard normal pseudorandom numbers and 𝑟 is specified so that the correlation between any two explanatory variables is given by 𝑟².

And observations on the dependent variable are then produced by 𝑦_𝑖=𝛽₁𝑥_𝑖1+𝛽₂𝑥_𝑖2+𝛽₃𝑥_𝑖3+𝛽₄𝑥_𝑖4+𝜀_𝑖,𝜀_𝑖∼𝑁0,𝜎²,𝑦_𝑖=𝛽₁𝑥_𝑖1+𝛽₂𝑥_𝑖2+𝛽₃𝑥_𝑖3+𝛽₄𝑥_𝑖4+𝛽₅𝑥_𝑖5+𝛽₆𝑥_𝑖6+𝜀_𝑖,𝜀_𝑖∼𝑁0,𝜎². (41)

In this paper we consider 𝑛=25,50, 𝑝=4,6, 𝜎²=0.1,0.25, and 𝑟=0.9,0.99,0.999. The simulation study results are given in Tables 1, 2, 3, 4, 5, 6, 7, and 8. By Tables 1–8, we can conclude that (1) when multicollinearity is serve, our new estimator performs well; (2) when 𝜎² is small, our new estimator performs well; (3) when 𝑛 is small, our new estimator performs well; (4) when 𝑝 is big, our new estimator performs well; (5) in all cases, our new estimator is better than the OLS estimator. So we can see that our new estimator not only is unbiased, but also can overcome multicollinearity. Our estimator is meaningful in practice.

Table 1 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=25, 𝑝=4, and 𝜎²=0.1.

Table 2 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=25, 𝑝=4, and 𝜎²=0.25.

Table 3 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=50, 𝑝=4, and 𝜎²=0.1.

Table 4 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=50, 𝑝=4, and 𝜎²=0.25.

Table 5 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=25, 𝑝=6, and 𝜎²=0.1.

Table 6 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=25, 𝑝=6, and 𝜎²=0.25.

Table 7 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=50, 𝑝=6, and 𝜎²=0.1.

Table 8 

Estimated MSE values of OLS, TP, AUTP, and UTP when 𝑛=50, 𝑝=6, and 𝜎²=0.25.

5. Conclusion

In this paper, we introduce an unbiased two-parameter estimator with prior information. We also show the superiority of the new estimator over the OLS estimator, the TP estimator, and the AUTP estimator in the MMSE sense. Furthermore, the estimators of the biasing parameters are also discussed in this paper.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation Project of CQ CSTC (Grant no. cstc2014jcyjA0999), the Scientific Research Foundation of Chongqing University of Arts and Sciences (Grant no. R2013SC12), and the National Natural Science Foundation of China (Grant no. 11201505).