Bi-log-concave distribution functions

Dümbgen, Lutz; Kolesnyk, Petro; Wilke, Ralf A. (2017). Bi-log-concave distribution functions. Journal of statistical planning and inference, 184, pp. 1-17. Elsevier 10.1016/j.jspi.2016.10.005

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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.

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