## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 55, 2007, 1-31

### A Kiefer-Wolfowitz theorem for convex densities

Fadoua Balabdaoui and Jon A. Wellner

#### Abstract

Kiefer and Wolfowitz showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum Likelihood Estimator $\widehat{F}_n$, which is, in fact, the least concave majorant of the empirical distribution function $\FF_n$, differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1} \log n)^{2/3}$ almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions $F$ with convex decreasing densities $f$, but with the maximum likelihood estimator $\widehat{F}_n$ of $F$ replaced by the least squares estimator $\widetilde{F}_n$: if $X_1 , \ldots , X_n$ are sampled from a distribution function $F$ with strictly convex density $f$, then the least squares estimator $\widetilde{F}_n$ of $F$ and the empirical distribution function $\FF_n$ differ in the uniform norm by no more than a constant times $(n^{-1} \log n )^{3/5}$ almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall, Hall and Meyer. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor.

#### Chapter information

**Source***Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007)

**Dates**

First available in Project Euclid: 4 December 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196797066

**Digital Object Identifier**

doi:10.1214/074921707000000256

**Mathematical Reviews number (MathSciNet)**

MR2435342

**Zentralblatt MATH identifier**

1176.62051

**Subjects**

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties

Secondary: 62G30: Order statistics; empirical distribution functions

**Keywords**

Brownian bridge convex density distance empirical distribution invelope process monotone density optimality theory shape constraints

**Rights**

Copyright © 2007, Institute of Mathematical Statistics

#### Citation

Balabdaoui, Fadoua; Wellner, Jon A. A Kiefer-Wolfowitz theorem for convex densities. Asymptotics: Particles, Processes and Inverse Problems, 1--31, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000256. https://projecteuclid.org/euclid.lnms/1196797066