Tests for monotonicity of a regression mean with guaranteed level

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Testing for stochastic monotonicity (2006, unknown): “(14) Sn = sup (y,x)∈Y×X Ûn(y,x) ĉn(x) ,”
TESTING REGRESSION MONOTONICITY IN ECONOMETRIC MODELS (2012, background): “Hall and Heckman (2000) (from now on, HH) developed a test based on the slopes of local linear estimates of f(·).”
Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function (2005, methodology): “The decreasing linear function F2(x) = −ax has already been studied by Gijbels et al. (2000) with a = 3σ. They get an estimated power of 77%. Gijbels et al. (2000) studied the function 0.”