# Test of Monotonicity by U-processes

##### Posted on Apr 23, 2022

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 $(-1, 1)$.

$U_n(t) = \frac{2}{n(n-1)} \sum_{1\le i < j \le n}\sign(Y_j-Y_i)\sign(X_i-X_j)k_n(X_i-t)k_n(X_j-t)$

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