2015/08/31 by Lutz Duembgen, Duembgen, Lutz, Petro Kolesnyk +3
Computer Science · Mathematics · #62G15 #62G30 #Advanced Statistical Methods and Models #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #Methodology (stat.ME) #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1508.07825
openalex publication_date 2015/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.