_{1}

This paper considers the asymptotic efficiency of the maximum likelihood estimator (MLE) for the Box-Cox transformation model with heteroscedastic disturbances. The MLE under the normality assumption (BC MLE) is a consistent and asymptotically efficient estimator if the “small ” condition is satisfied and the number of parameters is finite. However, the BC MLE cannot be asymptotically efficient and its rate of convergence is slower than ordinal order when the number of parameters goes to infinity. Anew consistent estimator of order is proposed. One important implication of this study is that estimation methods should be carefully chosen when the model contains many parameters in actual empirical studies.

The Box-Cox transformation model (BC model) [

It is well known that the MLE is usually an asymptotically efficient estimator when the number of parameters is finite. However, this may not be true when the number of parameters goes to infinity. It is often necessary for us to consider cases in which numbers of groups go to infinity. For example, the new medical payment system known as the Diagnostic Procedure Combination/Per Diem Payment System (DPC/PDPS) was introduced in 2003 in Japan, and as of April 2014, 1863 hospitals had either already joined or were preparing to join; this number has been increasing [

This paper considers the estimation of the Box-Cox transformation model with heteroscedastic disturbances when the number of groups that increases to infinity. In such cases, the conventional maximum likelihood method yields only an estimator whose rate of convergence is slower than ordinal order of

Suppose that

with heteroscedastic disturbances and variances given by

^{1}If the “small

where ^{1} Under this condition, we can assume that

where

Let the numbers of observations be

As before, although the values of derivatives are at

Here,

We get

Therefore, if

However, for the transformation parameter

we get [

under the “small

Since

we get

and (4) is not satisfied. This means that the MLE becomes a consistent estimator only of order

Therefore,

Here, an alternative estimator is proposed by an essential modification of the likelihood function. Suppose that disturbances are homoscedastic and that

where

Instead of maximizing (15), we considered the roots of the equations,

For the standard maximum likelihood method, the variance is estimated by the simple average. However, in this case, the variance is estimated by the weighted average of least squares residuals.

We assume

where

Here,

where

From (21), we get

where

where

This paper considers the estimation of the BC model with heteroscedastic disturbances; that is, variances are different by groups. The BC MLE is a consistent and asymptotically efficient estimator if the “small

The author would like to thank two anonymous referees for their helpful comments and suggestions.

Nawata, K. (2016) Asymptotic Efficiency of the Maximum Likelihood Estimator for the Box-Cox Transformation Model with Heteroscedastic Disturbances. Open Journal of Statistics, 6, 835-841. http://dx.doi.org/10.4236/ojs.2016.65069