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The aim of this paper is to propose some diagnostic methods in stochastic restricted linear regression models. A review of stochastic restricted linear regression models is given. For the model, this paper studies the method and application of the diagnostic mostly. Firstly, review the estimators of this model. Secondly, show that the case deletion model is equivalent to the mean shift outlier model for diagnostic purpose. Then, some diagnostic statistics are given. At last, example is given to illustrate our results.

In a linear regression, the ordinary least squares estimator (LS) is unbiased and has minimum variance among all linear unbiased estimators and has been treated as the best estimator for a long time. When the addition of stochastic linear restrictions on the unknown parameter vector was assumed to be held, Theil [

Nearly forty years, the diagnosis and influence analysis of linear regression model has been fully developed (R.D. Cook and S. Weisberg [

However, statistical diagnostics of stochastic restricted linear regression models based on stochastic restricted ridge estimator (SRRE) are studied in this paper. The paper is organized as follows. The model and the estimators are reviewed in Section 2. We show that the case deletion model is equivalent to the mean shift outlier model for diagnostic purpose in Section 3. Some diagnostic statistics are given in Section 4. The example to illustrate our results is given in Section 5.

Consider the following linear model:

where

Suppose that

where

Using the mixed approach, Durbin [

The mixed estimator is an unbiased estimator. However, when multicollinearity exists, the mixed estimator is no longer a good estimator.

Ozkale [

The result from the simulation study shows that SRRE outperform ME (see Wu and Liu [

The most classical ridge estimator for linear regression is the following:

proposed by Hoerl and Kennard [

an alternative of the estimator of

In Schaefer, et al. [

In Kibria, et al. [

This paper selects

Consider the stochastic restricted linear model, where the

This model is called case-deletion model. Supposed that the SRRE of the coefficient function

In order to study the influence of the

Theorem 1. For model (5), the SRRE of

and

where

Proof: Let

Supposed that

which leads to (6).

Because

and

hence

The other common statistical diagnosis model is the mean shift outlier model (MSOM). For the stochastic restricted linear regression model, the corresponding MSOM is

where the parameter

where

Theorem 2. For model (8), there are

Proof: By the matrix form of model (8), we obtained

On the other hand, by the formula of calculating the inverse matrix of partitioned matrix, we have

which leads to

Let

Theorem 3. Supposed that

where

Proof: Because

Substituting these results into (9) gives

W-K statistic is advanced from the view of data fitting. Considering the influence of the

which measure the influence of the

In order to illustrate the validity of above results, extensive Monte Carlo sampling experiments were conducted. To evaluate the finite-sample performance of our proposed method, we simulate 60 random samples from the following model:

The stochastic restricts as follows:

where

dology, we change the value of the first, 125th and 374th data. For every case, it is easy to obtain

From the

In this paper, stochastic restricted linear regression models are revisited. Useful diagnostic methods are derived. Through simulation study, we illustrate that our proposed methods can work fairly well.