## Electronic Journal of Statistics

### An improved global risk bound in concave regression

Sabyasachi Chatterjee

#### Abstract

A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator. The best known risk bound, as had appeared in Guntuboyina and Sen, scaled like $\log(en)\:n^{-4/5}$ under the mean squared error loss, up to a constant factor. The authors in [8] had conjectured that the logarithmic term may be an artifact of their proof. We show that indeed the logarithmic term is unnecessary and prove a risk bound which scales like $n^{-4/5}$ up to constant factors. Our proof technique has one extra peeling step than in a usual chaining type argument. Our risk bound holds in expectation as well as with high probability and also extends to the case of model misspecification, where the true function may not be concave.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 1608-1629.

Dates
Received: May 2016
First available in Project Euclid: 18 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1468847265

Digital Object Identifier
doi:10.1214/16-EJS1151

Mathematical Reviews number (MathSciNet)
MR3522655

Zentralblatt MATH identifier
1349.62126

#### Citation

Chatterjee, Sabyasachi. An improved global risk bound in concave regression. Electron. J. Statist. 10 (2016), no. 1, 1608--1629. doi:10.1214/16-EJS1151. https://projecteuclid.org/euclid.ejs/1468847265

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