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.