To overcome the multicollinearity problem in the linear regression model, several biased estimators were defined in the place of Ordinary Least Squares Estimator (OLSE) to estimate the regression coefficients, and the properties of these were discussed in the literature. Some of these estimators are based on only sample information such as Ridge Estimator (RE) [1] , Almost Unbiased Ridge Estimator (AURE) [2] , Liu Estimator (LE) [3] and Almost Unbiased Liu Estimator (AULE) [4] . However for each case, the researcher has to obtain their properties and to compare the superiority of one estimator over another estimator when selecting a suitable estimator for a practical situation. Therefore [5] proposed a Generalized Unrestricted Estimator (GURE) to represent the RE, AURE, LE and AULE, where is the OLSE, and A is a positive definite matrix which depends on the corresponding estimators of RE, AURE, LE and AULE.

However, the researchers are still trying to find the best estimator by changing the matrix A compared to the already proposed estimators based on sample information. Instead of changing A, in this research we introduce a more efficient new biased estimator based on optimal choice of A.

The rest of the paper is organized as follows. The model specification and estimation is given in Section 2. In Section 3, we propose a biased estimator namely Generalized Optimal Estimator (GOE), and we obtain its stochastic properties. In Section 4 we compare the proposed estimator with some biased estimators in the Scalar Mean Square Error criterion by using a real data set and a Monte Carlo simulation. Finally some conclusion remarks are given in Section 5.

First we consider the multiple linear regression model

where y is an observable random vector, X is an known design matrix of rank p, is a vector of unknown parameters and ε is an vector of disturbances.

The Ordinary Least Square Estimator of and are given by

and

respectively, where.

The Ridge Estimator (RE), Almost Unbiased Ridge Estimator (AURE), Liu Estimator (LE) and Almost Unbiased Liu Estimator (AULE) are some of the biased estimators proposed to solve the multicollinearity problem which are based only on sample information. The estimators are given below:

RE: where for

LE: where for

AURE: where for

AULE: where for

Since RE, LE, AURE and AULE are based on OLSE, [5] proposed a generalized form to represent these four estimators, the Generalized Unrestricted Estimator (GURE) which is given as

where is a positive definite matrix and stands for w, , and .

The bias vector, dispersion matrix and MSE matrix of are given as

and

respectively.

Instead of changing the matrix to introduce a new biased estimator, in this research we obtain the optimal choice of by minimizing the Mean Square Error Matrix (MSEM) of GURE with respect to.

From (7) the following equation can be obtained by taking trace operator as

By minimizing (8) with respect to, the optimum can be obtained.

where

Now we can simplify the matrix B as

Therefore

Now we will use the following three results (see [6] , p. 521, 522) to obtain the and.

(a) Let M and X be any two matrixes with proper order. Then

(b) If x is an n-vector, y is an m-vector, and C an matrix, then

(c) Let x be a K vector, M a symmetric matrix, and C a matrix. Then

By applying (a) we can obtain

Now we consider

By using (b) and (c) we can obtain

and

respectively.

Hence

Substituting (11) and (15) to (9), we can derive that

Equating (16) to a null matrix, we can obtain the optimal matrix as

Note that exists since (see [6] , p. 494).

Now we are ready to propose a biased estimator namely Generalized Optimal Estimator (GOE) as

The bias vector, dispersion matrix, mean square error matrix and scalar mean square error of can be obtained as

and

respectively.

Note that since is symmetric and where it can be defined. Then and. Therefore r is symmetric and idempotent

matrix. Now we write the optimal matrix as.

Now the bias vector, dispersion matrix, mean square error matrix and scalar mean square error of can be rewritten as

and

respectively.

For practical situations we have to replace the unknown parameters and. For an estimated value for the OLSE, RE, LE, AURE or AULE can be used. For the estimated value for we can use either (2) or replace in (2) by RE, LE, AURE or AULE accordingly. In the next section we will discuss the superiority of estimators when replacing each of these estimated values by using a simulation study, and then we use numerical examples for further illustration.

In this paper we proposed a new biased estimator namely Generalized Optimal Estimator (GOE) in a multiple linear regression when there exists multicollinearity problem in the independent variables. The proposed estimator is superior to biased estimators which are based on sample information and takes the form. Based on Tables 1-3, it can be concluded that the proposed estimator has smallest scalar mean square error values com- pared with RE, LE, AURE and AULE. We can also suggest that GOE-OLSE is the best estimator with compared to GOE-RE, GOE-LE, GOE-AURE and GOE-AULE.

We thank the Postgraduate Institute of Science, University of Peradeniya, Sri Lanka for providing all facilities to do this research.

SivarajahArumairajan,PushpakanthieWijekoon,11, (2015) Optimal Generalized Biased Estimator in Linear Regression Model. Open Journal of Statistics,05,403-411. doi: 10.4236/ojs.2015.55042