Nonparametric Inference About a Density's Mode via Shape Constraints

• Activity Number: 202
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Charles Doss*+ and Jon Wellner
• Companies: University of Minnesota and University of Washington
• Keywords: the mode ; log-concave ; likelihood ratio test ; shape constraints ; limiting distribution ; s-concave

Abstract

We consider inference about the mode of a density on the real line via the shape constraint of s-concavity, for -1 < s \le 0, with focus on the case s=0, which is log-concavity. The s-concave classes are all unimodal classes, so the mode is a natural parameter of interest. In nonparametric settings the mode is generally not estimable at a root-n rate and does not always have a normal limiting distribution, and current methods for testing or forming confidence intervals for the location of the mode are generally complicated. We construct a likelihood ratio test for the location of the mode. The test can be inverted to form a confidence interval. One benefit of likelihood ratio statistics is that their distributions are often asymptotically free of nuisance parameters. Our statistic has this property. We show this by decomposing our statistic into a sum of two terms, one depending on behavior local to the mode, and the other on behavior away from the mode, and seeing that the former yields the limit distribution while the latter is a negligible remainder term. This is joint work with Jon Wellner at the University of Washington.