## The Annals of Statistics

### The Penalty for Assuming that a Monotone Regression is Linear

#### Abstract

For jointly distributed random variables $(X, Y)$ having marginal distributions $F$ and $G$ with finite second moments and $F$ continuous, the proportion of $\operatorname{Var}(Y)$ explained by linear regression is $\lbrack\operatorname{Corr}(X, Y)\rbrack^2$ while the proportion explained by $E(Y \mid X)$ can be arbitrarily near 1. However, if the true regression, $E(Y\mid X)$, is monotone, then the proportion of $\operatorname{Var}(Y)$ it explains is at most $\operatorname{Corr}\lbrack Y, G^{-1}(F(X))\rbrack$.

#### Article information

Source
Ann. Statist., Volume 15, Number 1 (1987), 443-448.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176350279

Digital Object Identifier
doi:10.1214/aos/1176350279

Mathematical Reviews number (MathSciNet)
MR885750

Zentralblatt MATH identifier
0613.62090

JSTOR