Central limit theorems for the Lₚ-error of smooth isotonic estimators

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We investigate the asymptotic behavior of the Lₚ-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the Lₚ-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the L₂-distance between the kernel estimator and the smoothed Grenander-type estimator.