Estimating a Smooth Monotone Regression Function

The problem of estimating a smooth monotone regression function m will be studied. We will consider the estimator mSI consisting of a smoothing step (application of a kernel estimator based on a kernel K) and of a isotonisation step (application of the pool adjacent violator algorithm). The estimator mSI will be compared with the estimator mIS where these two steps are interchanged. A higher order stochastic expansion of these estimators will be given which show that mSI and mSI are asymptotically first order equivalent and that mIS has a smaller mean squared error than mSI if and only if the kernel function of the kernel estimator is not too smooth.