Optimal global rates of convergence for nonparametric regression with unbounded data

Estimation of regression functions from independent and identically distributed data is considered. The L2L2 error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates a boundedness assumption on the explanatory variable XX is made besides smoothness assumptions on the regression function and moment conditions on the response variable YY. In this article we consider the kernel estimate and show that by replacing the boundedness assumption on XX by a proper moment condition the same (optimal) rate of convergence can be shown as for bounded data. This answers Question 1 in Stone [1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10, 1040–1053].