Weak convergence of the least concave majorant of estimators for a concave distribution function

Brendan K Beare,Zheng Fang

doi:10.1214/17-ejs1349

Abstract

We study the asymptotic behavior of the least concave majorant of an estimator of a concave distribution function under general conditions. The true concave distribution function is permitted to violate strict concavity, so that the empirical distribution function and its least concave majorant are not asymptotically equivalent. Our results are proved by demonstrating the Hadamard directional differentiability of the least concave majorant operator. Standard approaches to bootstrapping fail to deliver valid inference when the true distribution function is not strictly concave. While the rescaled bootstrap of Dümbgen delivers asymptotically valid inference, its performance in small samples can be poor, and depends upon the selection of a tuning parameter. We show that two alternative bootstrap procedures—one obtained by approximating a conservative upper bound, the other by resampling from the Grenander estimator—can be used to construct reliable confidence bands for the true distribution. Some related results on isotonic regression are provided.

Highlights

Nonparametric estimation under shape constraints such as monotonicity and concavity has received increasing attention in recent years
As pointed out by Walther (2009), nonparametric estimation under shape constraints is attractive for two main reasons: (1) shape constraints are often implied by theoretical models or are at least plausible assumptions, and (2) nonparametric estimation under shape constraints is often feasible without the use of tuning parameters, as opposed to classical kernel or series estimators
Grenander (1956) showed that, given a random sample drawn from a nonincreasing probability density, the left-derivative of the least concave majorant (LCM) of the empirical distribution function achieves the maximum likelihood among all nonincreasing densities

Summary

Introduction

Nonparametric estimation under shape constraints such as monotonicity and concavity has received increasing attention in recent years. Have been provided by Groeneboom (1985), Groeneboom et al (1999), Kulikov and Lopuhaa (2005), Durot (2007) and Durot et al (2012) These global results require f to be strictly decreasing on its support. Our proof exploits the fact that the LCM operator is Hadamard directionally differentiable (see Definition 2.2 and Proposition 2.1 below) despite not being fully Hadamard differentiable This provides enough structure to invoke the Delta method (Shapiro, 1991; Dumbgen, 1993) and in this way derive the weak limit of Gn. Our result applies to the LCM of an empirical distribution function, but to any√estimator of F obtained by taking the LCM of an estimator Fn of F for which n(Fn − F ) converges weakly to a continuous process G vanishing at infinity. We denote by · the uniform norm on ∞(T ), where T should be clear from context

Weak convergence of Gn to G

Bootstrap approximation of the law of Gn

Uniform confidence bands for F

Proofs