One-Sided Inference about Functionals of a Density

Abstract

This paper discusses the possibility of truly nonparametric inference about functionals of an unknown density. Examples considered include: discrete functionals, such as the number of modes of a density and the number of terms in the true model; and continuous functionals, such as the optimal bandwidth for kernel density estimates or the widths of confidence intervals for adaptive location estimators. For such functionals it is not generally possible to make two-sided nonparametric confidence statements. However, one-sided nonparametric confidence statements are possible: e.g., "I say with 95% confidence that the underlying distribution has at least three modes." Roughly, this is because the functionals of interest are semicontinuous with respect to the topology induced by a distribution-free metric. Then a neighborhood procedure can be used. The procedure is to find the minimum value of the functional over a neighborhood of the empirical distribution in function space. If this neighborhood is a nonparametric $1 - \alpha$ confidence region for the true distribution, the resulting minimum value lowerbounds the true value with a probability of at least $1 - \alpha$. This lower bound has good asymptotic properties in the high-confidence setting $\alpha$ close to 0.