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Asymptotic tail bounds for the Dempfle-Stute estimator in general regression models. (English) Zbl 1384.62136
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 129-156 (2017).
Summary: 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 [A. Dempfle and W. Stute, Stat. Neerl. 56, No. 2, 233–242 (2002; Zbl 1076.62520)] 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 W. Stute and J. L. Wang [Ann. Stat. 21, No. 3, 1591–1607 (1993; Zbl 0785.60020)] on the conditional distribution of concomitants given the order statistics.
For the entire collection see [Zbl 1383.62010].
##### MSC:
 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference
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