Shape constrained nonparametric estimation

Abstract

Shape constrained nonparametric estimation dates back in the 1950's. In his milestone paper in 1956, Grenander found the maximum likelihood (ML) estimator of a nonincreasing density, whereas Brunk (1958) obtained the least squares estimator of a monotone regression function. After the derivation of these estimators, it has taken some time before distribution theory for such estimators entered the literature. Prakasa Rao (1969, 1970) established the limiting distribution of Grenander's estimator and of the ML estimator a monotone failure rate at a fixed point in the interior of the support. Typically, this involves convergence rates different from the traditional central limit theorem as well as a non-normal limit distribution, characterized by two-sided Brownian motion with a parabolic drift. Since then, similar estimators have been proposed in other statistical models, but despite the high interest and applicability, the difficulty in deriving of the distributional properties remained a major drawback. Shape constrained estimation was revived by Groeneboom (1985), who proposed an alternative for Prakasa Rao's bothersome type of proof. Groeneboom's approach employs a so-called inverse process, which has become a cornerstone in studying pointwise limit behaviour of several shape constrained nonparametric estimators. The last decade, this approach has turned out to be flexible enough also to handle global limit behaviour such as the L_k distance between the nonparametric estimator and the infinite dimensional parameter of interest. Part of my research interests is concerned with the limit behaviour of shape constrained nonparametric estimators. In this talk I will give an overview of some the results in this area and discuss some recent developments. This includes the derivation of the limit behaviour of the supremum distance, which solves a problem that has been open for more than twenty years.