## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 55, 2007, 85-100

### Confidence bands for convex median curves using sign-tests

#### Abstract

Suppose that one observes pairs $(x_1,Y_1)$, $(x_2,Y_2)$, \ldots, $(x_n,Y_n)$, where $x_1 \le x_2 \le \cdots \le x_n$ are fixed numbers, and $Y_1, Y_2, \ldots, Y_n$ are independent random variables with unknown distributions. The only assumption is that ${\rm Median}(Y_i) = f(x_i)$ for some unknown convex function $f$. We present a confidence band for this regression function $f$ using suitable multiscale sign-tests. While the exact computation of this band requires $O(n^4)$ steps, good approximations can be obtained in $O(n^2)$ steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size $n$ tends to infinity.

#### 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/1196797069

**Digital Object Identifier**

doi:10.1214/074921707000000283

**Zentralblatt MATH identifier**

1176.62047

**Subjects**

Primary: 62G08: Nonparametric regression 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

Secondary: 62G35: Robustness

**Keywords**

computational complexity convexity distribution-free pool-adjacentviolators algorithm Rademacher variables signs of residuals

**Rights**

Copyright © 2007, Institute of Mathematical Statistics

#### Citation

Dümbgen, Lutz. Confidence bands for convex median curves using sign-tests. Asymptotics: Particles, Processes and Inverse Problems, 85--100, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000283. https://projecteuclid.org/euclid.lnms/1196797069