Bi-log-concave Distribution Functions

Lutz Dümbgen, Petro Kolesnyk, Ralf Wilke

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Abstract

Nonparametric statistics for distribution functions F or densities f=F′ under qualitative shape constraints constitutes 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 logF 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 log-concavity of f, bi-log-concavity of F allows for multimodal densities. We provide various characterisations. 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.
Original languageEnglish
JournalJournal of Statistical Planning and Inference
Volume184
Pages (from-to)1-17
Number of pages17
ISSN0378-3758
DOIs
Publication statusPublished - May 2017

Bibliographical note

Published online: 23. November 2016

Keywords

  • Hazard
  • Honest confidence region
  • Moment generating function
  • Moments
  • Reverse hazard
  • Shape constraint

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