Confidence bands for densities, logarithmic point of view

Abstract

Let $f$ be a probability density and $C$ be an interval on which $f$ is bounded away from zero. By establishing the limiting distribution of the uniform error of the kernel estimates $f_n$ of $f$, Bickel and Rosenblatt (1973) provide confidence bands $B_n$ for $f$ on $C$ with asymptotic level $1-\alpha\in]0,1[$. Each of the confidence intervals whose union gives $B_n$ has an asymptotic level equal to one; pointwise moderate deviations principles allow to prove that all these intervals share the same logarithmic asymptotic level. Now, as soon as both pointwise and uniform moderate deviations principles for $f_n$ exist, they share the same asymptotics. Taking this observation as a starting point, we present a new approach for the construction of confidence bands for $f$, based on the use of moderate deviations principles. The advantages of this approach are the following: (i) it enables to construct confidence bands, which have the same width (or even a smaller width) as the confidence bands provided by Bickel and Rosenblatt (1973), but which have a better aymptotic level; (ii) any confidence band constructed in that way shares the same logarithmic asymptotic level as all the confidence intervals, which make up this confidence band; (iii) it allows to deal with all the dimensions in the same way; (iv) it enables to sort out the problem of providing confidence bands for $f$ on compact sets on which $f$ vanishes (or on all $\bb R^d$), by introducing a truncating operation.