Test of Monotonicity by U-processes
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This note is for Ghosal, S., Sen, A., & van der Vaart, A. W. (2000). Testing Monotonicity of Regression. The Annals of Statistics, 28(4), 1054–1082.
They determine the critical regions of the tests by computing the limiting distributions of the test statistics with the help of the empirical process approximation of the U-process.
Consider testing the hypothesis,
\[H_0: m(\cdot) \text{ is an increasing function on $T$}\]Suppose $m(\cdot)$ is continuously differentiable. Then
\[H_0: m'(t) \ge 0 \text{ for all $t\in T$}\]Let $k(\cdot)$ be a nonnegative, symmetric, continuous kernel supported in $[-1, 1]$ and twice continuously differentiable in $(-1, 1)$.
\[U_n(t) = \frac{2}{n(n-1)} \sum_{1\le i < j \le n}\sign(Y_j-Y_i)\sign(X_i-X_j)k_n(X_i-t)k_n(X_j-t)\]