Choosing a Density Estimate

Abstract

Consider the following simple situation: g _n and f _n are two density estimates, and we must select the best one, that is, arg min(∫ |f _n − f|, ∫ |g _n − f|). More precisely, given the sample X ₁, …, X _n distributed according to density f, we are asked to construct a density estimate φ _n such that

$$ \int {|{\varphi _n} - f| \approx {\text{min}}\left( {\int {|fn - f|,\int {|{g_n} - f|} } } \right).} $$

This simple problem turns out to be surprisingly difficult, even if the estimates f _n and g _n are fixed densities, not depending on the data.