In case of misspecification, the Akaike information
criterion (AIC; Akaike, 1973, in Petrov & Csaki, eds.,
Second International Symposium on Information Theory,
pp. 267–281. Budapest: Akademia Kiado) is an asymptotically
biased estimator of the expected Kullback–Leibler
discrepancy. This paper gives simple expressions for the
bias that can be used to construct improved estimators.
However, for the examples that are considered in detail
it turns out that model selection procedures based on such
improved estimators are nearly equivalent to model selection
procedures based on severely biased estimators.