Test of Monotonicity by Calibrating for Linear Functions

Posted on May 11, 2022 (Update: Jun 14, 2022)

Tags: Monotone Function

This note is for Hall, P., & Heckman, N. E. (2000). Testing for Monotonicity of a Regression Mean by Calibrating for Linear Functions. The Annals of Statistics, 28(1), 20–39.

The bandwidth test will generally perform well if the regression function does not have any flat or nearly flat parts,

Let $0 \le r\le s-2\le n-2$ be integers, let $a, b$ be constants and put

\[S(a, b\mid r,s) = \sum_{i=r+1}^s[Y_i-(a+bx_i)]^2\]

For each choice of $(r, s)$, define $\hat a = \hat a(r, s)$ and $\hat b = \hat b(r, s)$ by

\[(\hat a, \hat b) = \argmin_{(a, b)} S(a, b\mid r, s)\]