Estimating Nonsmooth Kímodal Densities and Their Inaeection Points

In literature several estimates have been proposed for unimodal densities. They are typically derived from the Grenander estimate (see Grenander (1956)) for decreasing densities which can be easily extended to the case of unimodal densities with a known mode at . The resulting estimator is the nonparametric maximum likelihood estimator (NPMLE). If the mode is unknown, the NPMLE does not exist anymore. One possible solution that has been proposed by Wegman (1969, 1970a,b) is to add the additional constraint of a modal interval of length , where has to be chosen by the statistician. More recently Bickel and Fan (1996) and Birg e (1997) proposed methods that are based on an initial mode estimate ̂ and an application of the NPMLE with mode ̂. The initial mode estimate requires the calculation of the MLE of f for O(n) candidate points for the mode, where n denotes the sample size. They showed their methods to provide good estimates both for nonsmooth densities and for the mode of nonsmooth densities.