Bi-log-concave distribution functions

TL;DR

This paper introduces the concept of bi-log-concave distribution functions, a new shape constraint that allows for multimodal densities and improves nonparametric confidence bands, especially in the tails.

Contribution

It defines bi-log-concavity for distribution functions, characterizes this class, and demonstrates how it enhances confidence band accuracy in nonparametric statistics.

Findings

Bi-log-concavity includes many common distributions.

Combining confidence bands with bi-log-concavity improves tail estimates.

Confidence bounds for moments and moment generating functions are derived.

Abstract

Nonparametric statistics for distribution functions F or densities f=F' under qualitative shape constraints provides an interesting alternative to classical parametric or entirely nonparametric approaches. We contribute to this area by considering a new shape constraint: F is said to be bi-log-concave, if both log(F) and log(1 - F) are concave. Many commonly considered distributions are compatible with this constraint. For instance, any c.d.f. F with log-concave density f = F' is bi-log-concave. But in contrast to the latter constraint, bi-log-concavity allows for multimodal densities. We provide various characterizations. It is shown that combining any nonparametric confidence band for F with the new shape-constraint leads to substantial improvements, particularly in the tails. To pinpoint this, we show that these confidence bands imply non-trivial confidence bounds for arbitrary moments and the moment generating function of F.