Asymptotic Tail Bounds for the Dempfle-Stute Estimator in General Regression Models

In a nonparametric regression model let the regression function m have a split-point $$\theta $$ , i.e., m runs above the mean output to the left of $$\theta $$ and it runs below that level to the right-hand side. Here, there can be a continuous crossing, but also an abrupt jump. We investigate an estimator which goes back to Dempfle and Stute (2002) in the special case that m is a unit step function with jump at $$\theta $$ . Under very mild local conditions on m we derive asymptotic tail bounds of integral-type. In particular, these bounds guarantee stochastic boundedness, which in turn is an essential part in deriving distributional convergence and corresponding extensions. Our proof relies on the Doob-Meyer decomposition of marked empirical distribution functions which enable us to apply a suitable martingale inequality. Moreover, we use a result of Stute and Wang (1993) on the conditional distribution of concomitants given the order statistics.