Abstract

Under a balanced loss function, we derive the explicit formulae of the risk of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the feasible minimum mean squared error (FMMSE) estimator, and the adjusted feasible minimum mean squared error (AFMMSE) estimator in a linear regression model with multivariate errors. The results show that the PSR estimator dominates the SR estimator under the balanced loss and multivariate errors. Also, our numerical results show that these estimators dominate the ordinary least squares (OLS) estimator when the weight of precision of estimation is larger than about half, and vice versa. Furthermore, the AFMMSE estimator dominates the PSR estimator in certain occasions.

1. Introduction

In the literature, many statisticians have studied the risk comparisons of various estimators in the linear model with normal errors and have generated substantial results. However, the assumption of normality restricts the range of possible applications. The multivariate distributions are more realistic and accurate than multivariate normal distributions in modeling real-word data due to their heavy tails. Moreover, multivariate distribution plays an important role in robust statistical inference. Therefore, various inference problems based on these distributions have been studied. The sampling performance of estimators is an important aspect among them.

Let us now consider a linear regression model where is an vector of observations on a dependent variable. is an full rank matrix of observations. is a vector of coefficients. We assume that has a multivariate distribution with the probability density function given by where . It is well known that its mean vector and covariance matrix are given by As is shown in Zellner [1], the multivariate distribution can be viewed as a mixture of multivariate normal and inverted gamma distributions: where

The ordinary least squares (OLS) estimator of is , where . Also, the Stein-rule (SR) estimator is where , , and is a constant such that . Under the mean squared error of prediction, Stein [2] and James and Stein [3] proved that the SR estimator dominates the OLS estimator when the numbers of explanatory variables are more than two and the MSE of the SR estimator is minimized if . Thus, we use this value of hereafter. From then on, lots of improved estimators have been proposed. For example, Baranchik [4] proposed the positive-part Stein-rule (PSR) estimator defined as Farebrother [5] proposed the feasible minimum mean squared error (FMMSE) estimator which is Further, Ohtani [6] extended the FMMSE estimator to the adjusted feasible minimum mean squared error (AFMMSE) estimator by adjusting the degrees of the freedom of the component of the FMMSE estimator. The AFMMSE estimator is

Some results related to the comparisons of these estimators have been established. For example, Giles [7] considered the pretest estimator for linear restrictions. Namba [8] studied the PMSE performance of the biased estimators in a regression model when relevant regressors are omitted. Namba and Ohtani [9] gave the risk comparison of the Stein-rule estimator under the Pitman nearness criterion. There is a common characteristic in their studies. That is, the used loss functions were the quadratic function and its variants. However, in regression analysis, we are often interested in using an estimator which has high precision of estimation and high goodness of fit of model. In this situation, Zellner [10] proposed a balanced loss function which takes account of both precision of estimation and goodness of fit. Balanced loss function is a more comprehensive and reasonable standard than quadratic loss and residual sum of squares. Much work has been done about the balanced loss risk comparisons of improved estimators in the normal linear model. Some examples are Giles et al. [11], Ohtani et al. [12], Ohtani [13], and so on. Their results show that SR estimator is not admissible and is dominated by PSR estimator. However, do the conclusions still hold under multivariate errors and balanced loss function? And, do these estimators still dominate the OLS estimator? It is interesting to discuss them under multivariate distributions and balanced loss function. Thus, we will give the explicit formulae for the balanced loss risk of these estimators and compare their sampling performance by theoretical and numerical analysis. In the next section, the explicit formulae of balanced loss risk of these estimators are derived. In Section 3, we compare the risk performance by numerical evaluations. The proofs of main results are given in Section 4.

2. Balanced Loss Function and Risk

In order to discuss the performance of considered estimators, we consider the balanced loss function as where is a scalar such that , and is any estimator of . The corresponding risk function is . Since has a multivariate distribution which can be viewed as the mixture of multivariate normal and inverted gamma distribution, we have

If the null hypothesis is and the alternative is , then the test statistic for is . In the same way as that of Namba [8], we consider the general pretest estimator as where is an indicator function such that if an event occurs and otherwise. is the critical value of the pretest, and is an arbitrary integer. The term reduces to the SR estimator when , and , and it reduces to the PSR estimator when , and . Furthermore, reduces to the FMMSE estimator when , , and , and it reduces to the AFMMSE estimator when , , and , respectively.

To derive the formulae of , we first compute , assuming that is given. If we denote , , then , and for given , where , is the noncentral chi-square distribution with degrees of freedom and noncentrality parameter . Thus, using and , we define the functions and as where are arbitrary integers. By direct computation, we have In the following, we first give one lemma in order to obtain the explicit formulae of risk.

Lemma 1. The explicit formulae of and are where , , and .

By this lemma and (11) and (14), we have the following theorem.

Theorem 2. Under model (1) and loss function (10), the risk of the general pretest estimator is

According to this theorem, we can obtain the risk of SR, PSR, FMMSE, and AFMMSE estimators, respectively, and discuss their dominance properties. Firstly, we analyze the dominance properties between SR and PSR estimators.

Theorem 3. The PSR estimator dominates the SR estimator in terms of balanced loss risk when the error term of the model obeys a multivariate distribution.

When the error term obeyed a multivariate normal distribution, Baranchik [4] proved that the PSR estimator dominated uniformly the SR estimator under the quadratic loss, and Ohtani [13] also proved that the SRSV estimator dominated uniformly the SR estimator under a balanced loss function. This theorem shows that when the loss function is extended to a balanced loss function, the dominance of the PSR estimator over the SR estimator still holds even if the error term obeys a multivariate distribution. This implies that the SR estimator is not admissible under a balanced loss function and multivariate errors.

Since further theoretical analysis of the risk of the SR, PSR, FMMSE, and AFMMSE estimators is difficult, we will compare them by numerical analysis in the next section.

3. Numerical Analysis

In order to compare the balanced loss risk of the estimators, we evaluated the values of relative risk defined by . Thus, the estimator has smaller risk than the OLS estimator when the value of relative risk is smaller than unity. By Theorem 2, we can obtain the risks of the OLS, SR, PSR, FMMSE, and AFMMSE estimators, respectively. In the following, taking , the parameter values used in the numerical evaluations are = various values, , , , and . The numerical evaluations are executed on a personal computer using Version 7.9 (R2009b) MATLAB Software. In order to evaluate the integral in the risk expressions of these estimators, we use Trapezoidal method with equal subdivisions. Following the method used by Namba [8], the infinite series in these risk expressions is judged to converge when the increment of the infinite series becomes smaller than . Now, we give the relative risk of the SR, PSR, FMMSE, and AFMMSE estimators for the case of , , , , and in Tables 1 and 2, respectively. According to Tables 1 and 2, it is sufficient to illustrate the result of Theorem 3. That is, the PSR estimator dominates the SR estimator under a balanced loss even if the error term obeys a multivariate distribution. We also find that when precision of estimation is more important (i.e., ), the SR and PSR estimators dominate the OLS estimator under the balanced loss function, and vice versa. This shows that the dominance of the SR and PSR estimators over the OLS estimator is not robust about the loss function. From Table 1, the FMMSE and AFMMSE estimators dominate the OLS estimator when the weight of precision of estimation is larger than about half, and vice versa. This indicates that the dominance results of the FMMSE and AFMMSE estimators over the OLS estimator do not hold necessarily under the balanced loss function. It is easy to see that the risk of the AFMMSE estimator is much smaller than the risks of the SR and PSR estimators if . However, the AFMMSE estimator does not dominate the FMMSE estimator under the balanced loss function when .

In sum, our results show that when the loss function and error terms are extended from the usual quadratic loss function and normal distribution to balanced loss function and multivariate distribution, the dominance of the PSR estimator over the SR estimator is robust. However, the dominance of these estimators over the OLS estimator is not robust.

4. Proof of Main Results

Proof of Lemma 1. For given , and ; meanwhile, and are mutually independent. Therefore, we have where is the region such that .
Making use of the change of variables, , , the integral in (17) reduces to Again, making use of the change of variables, , , the integral in (18) becomes Further, making use of the change of a variable, , the integral in (19) reduces to By (17)–(20), we have
Next, we derive the formula for . Noting that and differentiating with respect to , we have
Since and , can be expressed as where is the density function of and Differentiating (23) with respect to , we have which together with (22) yields that Multiplying from the left of the above, we have This completes the proof of this lemma.

Proof of Theorem 2. By Lemma 1, we have where . Making use of the change of a variable, , (28) becomes Taking , (29) becomes which together with (28) and (29) yields
In a similar way, we have Obviously, we have . This together with (11), (14), (31), and (32) yields the expression of . The proof of this theorem is completed.

Proof of Theorem 3. By Theorem 2, let ; we have
Since hence, differentiating (33) with respect to and performing some manipulations, we have From (35), when , a condition for to be monotonically decreasing is Thus, is monotonically decreasing on if . Since becomes the SR estimator when , and and it reduces to the PSR estimator when , , and , the PSR estimator dominates the SR estimator. This completes the proof.