BOOTSTRAPPING LOG LIKELIHOOD AND EIC, AN EXTENSION OF AIC

MAKIO ISHIGURO, YOSIYUKI SAKAMOTO AND GENSHIRO KITAGAWA

The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

(Received November 18, 1994; revised February 18, 1997)

Abstract. Akaike (1973, 2nd International Symposium on Information Theory, 267–281, Akademiai Kiado, Budapest) proposed AIC as an estimate of the expected log likelihood to evaluate the goodness of models fitted to a given set of data. The introduction of AIC has greatly widened the range of application of statistical methods. However, its limit lies in the point that it can be applied only to the cases where the parameter estimation are performed by the maximum likelihood method. The derivation of AIC is based on the assessment of the effect of data fluctuation through the asymptotic normality of MLE. In this paper we propose a new information criterion EIC which is constructed by employing the bootstrap method to simulate the data fluctuation. The new information criterion, EIC, is regarded as an extension of AIC. The performance of EIC is demonstrated by some numerical examples.

Key words and phrases: Log likelihood, AIC, bootstrap, MLE, information criterion, model selection, estimator selection, bias correction, expected log likelihood, penalized log likelihood, predictive distribution.