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In this paper we compare recently developed preliminary test estimator called Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE) with Ordinary Least Square Estimator (OLSE) and Mixed Estimator (ME) in the Mean Square Error Matrix (MSEM) sense for the two cases in which the stochastic restrictions are correct and not correct. Finally a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings.

To overcome the multicollinearity problem arises in the Ordinary Least Squares Estimation (OLSE) procedure, different methods have been proposed in the literature. One of the most important estimation methods is to consider biased estimators, such as the Ridge Estimator (RE) by Hoerl and Kennard [

In this research we further compare the mean square error matrix of PTSRLE with OLSE and ME. The rest of the paper is organized as follows. The model specification and estimation are given in section 2. In section 3, the mean square error matrix comparisons between PTSRLE with OLSE and ME are performed. A numerical example and a Monte Carlo simulation are used to illustrate the theoretical findings in section 4, and in section 5 we state the conclusions.

First we consider the multiple linear regression model

where y is an n × 1 observable random vector, X is an n × p known design matrix of rank p, β is a p × 1 vector of unknown parameters and ε is an n × 1 vector of disturbances.

In addition to sample model (1), let us be given some prior information about β in the form of a set of m independent stochastic linear restrictions as follows;

where r is an m × 1 stochastic known vector R is a m × p of full row rank

The Ordinary Least Squares Estimator (OLSE) for model (1) and the Mixed Estimator (ME) (Theil and Goldberger [

and

respectively, where

The expectation vector, and the mean square error matrix of

and

respectively.

The expectation vector, dispersion matrix, and the mean square error matrix of

and

respectively, where,

The Liu Estimator (Liu, [

Replacing OLSE by ME in the Liu Estimator, Hubert and Wijekoon [

when different estimators are available for the same parameter vector

where

Mean Square Error of

For two given estimators

Let us now turn to the question of the statistical evaluation of the compatibility of sample and stochastic information. The classical procedures is to test the hypothesis

under linear model (1) and stochastic prior information (2).

The Ordinary Stochastic Preliminary Test Estimator (OSPE) of

Further, we can write (15) as

where,

which has a non-central

and

scripted interval, and zero otherwise.

The expectation vector, dispersion matrix, and the mean square error matrix of

an

respectively, where,

Recently, Arumairajan and Wijekoon [

Note that the PTSRLE can be rewritten as follows:

By using Equations (19), (20) and (21), Arumairajan and Wijekoon [

and

respectively, where

In this section we compare the PTSRLE with OLSE and ME in the mean square error matrix sense for the two cases in which the stochastic restrictions are correct and not correct.

The mean square error matrix difference between OLSE and PTSRLE can be written as

where,

with

Now the following theorem can be stated

Theorem 1:

1) When the stochastic restrictions are true (i.e.

2) When the stochastic restrictions are not true (i.e.

Proof:

1) If stochastic restrictions are correct then

To apply lemma 2 (Appendix), we have to show that

Since

Hence

2) If stochastic restrictions are not correct then

We have already proved that

mum eigenvalue of

The mean square error matrix difference between ME and PTSRLE is

where

Now we can give the following theorem.

Theorem 2:

1) When the stochastic restrictions are true (i.e.

2) When the stochastic restrictions are not true (i.e.

Proof:

1) When stochastic restrictions are true then

2) When stochastic restrictions are not true (i.e.

According to theorem 1 and 2 it is clear that PTSRLE is superior to OLSE and ME under certain conditions.

In this section the comparison of PTSRLE with OLSE and ME are demonstrated using a numerical example, and a simulation study.

To illustrate our theoretical result, we consider the data set on Total National Research and Development Expenditures as a percent of Gross National product due to Gruber [

The four column of the 10 × 4 matrix X comprise the data on

1) The eigen values of

2) The OLS estimator of

3) The OLS estimator of

4) The condition number of

The condition number implies the existence of multicollinearity in the data set. We consider the following stochastic restrictions (Li and Yang, [

Further the significance level is taken as α = 0.05.

Based on

To illustrate the behavior of our proposed estimators, we perform the Monte Carlo Simulation study by considering different levels of multicollinearity. Following McDonald and Galarneau [

where

where

According to

In this paper we have shown that the Preliminary Test Stochastic Restricted Liu Estimator is superior to Mixed Estimator and Ordinary Least Square Estimator in the mean square error matrix sense under certain conditions.

From the simulation study and the numerical illustration we notice that the PTSRLE has the smallest SMSE than ME and OLSE when multicollinearity among the predictor variables is large.

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

SivarajahArumairajan,PushpakanthieWijekoon,11, (2015) More on the Preliminary Test Stochastic Restricted Liu Estimator in Linear Regression Model. Open Journal of Statistics,05,340-349. doi: 10.4236/ojs.2015.54035

Lemma 1: (Rao and Touterburg, [

Let A and B be (n × n) matrices such that A > 0 and

Lemma 2: (Farebrother, [

Let A > 0 be an (n × n) matrix, b an (n × 1) vector. Then

Lemma 3: (Wang et al., [

Let n × n matrices M > 0, N > 0 (or

Lemma 4: (Trenkler and Toutenburg, [

Let