On the asymptotic Mean Square Error of L1 kernel estimates of smooth functions

AbstractRates of convergence of Mean Squared Error of convolution type estimators of regression functions or density functions in E∞ are derived, employing L1 kernel functions. The idea is to let the order of the kernel (number of vanishing moments) tend to infinity with increasing number of observations. In this setting, the rate n−1 is achieved if and only if the function to be estimated has a specific property. For a broad class of functions, the optimal rate is seen to be O(αn12n), where (αne)xn ∼ n