A Note on the Modified Likelihood for Density Estimation

Abstract

Let f _λ be a kernel estimate (with window width λ) of the density f. Its performance is assessed by the Kullback-Leibler information distance I(f, f _λ) = ∫ f log f − ∫ f log f _λ. This article establishes conditions for the asymptotic equivalence of the cross-validation estimate and the jackknife estimate of the term ∫ f log f _λ, and provides the common limiting value. This gives insight into the “modified likelihood” criterion for choosing λ, introduced by Habbema, Hermans, and Van den Broek (1974) and Duin (1976).

Key Words:

• Cross-validation
• Kullback-Leibler information
• Modified likelihood
• Density estimation