## AbstractWe prove asymptotic normality for $L_k$-functionals $\int |\hat F_n-F_n|^k g(t)\,dt$, where $F_n$ is the empirical distribution function of a sample from a decreasing density and $\hat F_n$ is the least concave majorant of $F_n$. From this we derive two test statistics for the null hypothesis that a probability density is monotone. These tests are compared with existing proposals such as the supremum distance between $\hat F_n$ and $F_n$.
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