Bi-Log-Concave Distribution and Regression Functions

• Activity Number: 218
• Type: Invited
• Date/Time: Monday, August 10, 2015 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Lutz Duembgen* and Ralf Andreas Wilke and Petro Kolesnyk
• Companies: University of Bern and Copenhagen Business School and University of Bern
• Keywords: log-concavity ; confidence band ; logistic regression ; link function ; moments

Abstract

When estimating a probability density or regression function, it is well-known that shape constraints such as monotonicity or (log-)concavity often lead to good and adaptive estimators which don't require smoothing parameters to be chosen. In this talk I present a new shape constraint for distribution functions: A distribution function F is called bi-log-concave if both log(F) and log(1-F) are concave. A special case are distribution functions with log-concave density f = F', but the new constraint is much weaker and allows, for instance, for multimodal densities. It is shown that combining this shape constraint with known confidence bands leads to substantially improved confidence regions for F itself and functionals of F. In the context of binary regression, bi-log-concavity of a regression function may be viewed as a nonparametric extension of logistic regression.

This is joint work with Petro Kolesnyk (Bern) and Ralf Wilke (Copenhagen).