Nonparametric Maximum Likelihood Estimation of a Log-Concave Density Based on Censored Data

• Activity Number: 296
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Dominic Schuhmacher*+ and Lutz Duembgen and Kaspar Rufibach
• Companies: University of Goettingen and University of Bern and Roche Biostatistics Oncology
• Keywords: censoring ; nonparametric MLE ; log-concave density ; EM algorithm ; survival analysis

Abstract

Event-times are often not observed exactly, but only after some form of censoring took place. This may be any combination of rounding/binning, right- and interval-censoring. Especially in medical applications it may furthermore be the case that the event in question does not occur for all subjects, e.g.\ some patients may be cured and will not die of the investigated disease while still being lost to follow-up. We study the nonparametric maximum likelihood estimator for such data, imposing a log-concavity constraint on its density. Log-concave densities cover a wide range of distributions with non-decreasing hazard rates. Also the constraint allows us to obtain an automatic, i.e.\ tuning-parameter free, smoothing of the classical unconstrained nonparametric estimators like the one by Kaplan and Meier. The estimator we use was proposed in a technical report by D\"umbgen, H\"usler, and Rufibach (2007) and an algorithm for its computation was sketched. The current work presents new theoretical findings, in particular existence, shape and consistency results for the estimator, and gives a refined version of the algorithm. An implementation is available in the R-package logconcens.