A Revision of &amp;lt;i&amp;gt;AIC&amp;lt;/i&amp;gt; for Normal Error Models

Kunio Takezawa

doi:10.4236/ojs.2012.23038

Abstract

Conventional Akaike’s Information Criterion (AIC) for normal error models uses the maximum-likelihood estimator of error variance. Other estimators of error variance, however, can be employed for defining AIC for normal error models. The maximization of the log-likelihood using an adjustable error variance in light of future data yields a revised version of AIC for normal error models. It also gives a new estimator of error variance, which will be called the “third variance”. If the model is described as a constant plus normal error, which is equivalent to fitting a normal distribution to one-dimensional data, the approximated value of the third variance is obtained by replacing (n-1) (n is the number of data) of the unbiased estimator of error variance with (n-4). The existence of the third variance is confirmed by a simple numerical simulation.

Highlights

Akaike’s Information Criterion (AIC) for multiple linear models with normal i.i.d. errors is defined as (e.g., [1,2])2l X, y aj, ˆ 2 2q 4, (1)where n is the number of data and q is the number of predictors of the multiple linear model
The maximization of the log-likelihood using an adjustable error variance in light of future data yields a revised version of AIC for normal error models. It gives a new estimator of error variance, which will be called the “third variance”
If the model is described as a constant plus normal error, which is equivalent to fitting a normal distribution to one-dimensional data, the approximated value of the third variance is obtained by replacing (n − 1) (n is the number of data) of the unbiased estimator of error variance with (n − 4)

Summary

Introduction

Akaike’s Information Criterion (AIC) for multiple linear models with normal i.i.d. errors is defined as (e.g., [1,2]). L X , y aj , ˆ 2 is the log-likeli- hood of the regression model in light of the data at hand ˆ 2 defined above is used as the error variance in AIC because AIC is a statistic based on the maximum-likelihood estimator. In this paper, we discusses the adjustment of error variance to calculate AIC for normal error models after recalling the derivation of conventional AIC for normal error models. This consideration leads to a new estimator of error variance, which will be called the “third variance”. The existence of the third variance is shown by a simple numerical simulation

Derivation of AIC for Normal Error

E RSS E y y t y y

Adjustment of Error Variance of AIC for Normal Error Models

Numerical Simulation

Conclusion