Nonparametric Inference About a Density's Mode via the Log-Concave Shape Constraint

• Activity Number: 424
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Charles Doss*+ and Jon Wellner
• Companies: University of Washington and University of Washington
• Keywords: the mode ; log-concave ; likelihood ratio test ; shape constraints ; the wilks phenomenon ; limiting distribution

Abstract

We consider the shape constraint of log-concavity in the setting of nonparametric density estimation. The constraint of log-concavity can serve as a surrogate for the constraint of unimodality, and 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 by comparing the log-concave maximum likelihood estimate (MLE) to the MLE over the constrained subclass composed of log-concave densities with a fixed mode. The test can be inverted to form a confidence interval. We study the properties of the constrained MLE and the Wilks phenomenon of the likelihood ratio statistic. This is joint work with Jon Wellner at the University of Washington.