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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].
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI
[1] Birnbaum, Z., Marshall A.: Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist. 32 , 687-703 (1961) · Zbl 0114.08004
[2] Bojanic, R., Seneta, E.: Slowly varying functions and asymptotic relations. J. Math. Anal. Appl. 34 , 302-315 (1971) · Zbl 0222.26003
[3] Dempfle, A., Stute, W.: Nonparametric estimation of a discontinuity in regression. Stat. Neerl. 56 , 233-242 (2002) · Zbl 1076.62520
[4] Ferger, D.: Stochastische Prozesse mit Strukturbrüchen. Dresdner Schriften zur Mathematischen Stochastik 7 , 1-123 (2009)
[5] Ferger, D., Klotsche, J., Lüken, U.: Estimation and testing of crossing-points in fixed design regression. Stat. Neerl. 66 , 380-402 (2012)
[6] Ferger, D.: Arginf-sets of multivariate cadlag processes and their convergence in hyperspace topologies. Theory Stoch. Process. 20 (36) , 13-41 (2015) · Zbl 1363.60050
[7] Ferger, D., Venz, J.: Density estimation via best · Zbl 1413.62048
[8] Knight, K.: Limiting distributions for · Zbl 0929.62021
[9] Massart, P.: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 , 1269-1283 (1990) · Zbl 0713.62021
[10] Matheron G.: Random Sets and Integral Geometry. John Wiley & Sons, New York, London, Sydney, Toronto (1975) · Zbl 0321.60009
[11] Molchanov I.: Theory of Random Sets. Springer-Verlag, London (2005) · Zbl 1109.60001
[12] Nguyen H.T.: An Introduction to Random Sets. Chapman & Hall/CRC, Boca Raton, London, New York (2006) · Zbl 1100.60001
[13] Shorack, G.R., Wellner, J.A.: Empirical Processes With Applications to Statistics. John Wiley & Sons, New York (1986) · Zbl 1170.62365
[14] Smirnov, N.V.: Limit distributions for the terms of a variational series. Amer. Math. Soc. Transl. Ser. (1) 11 , 82-143 (1952)
[15] Stute, W.: Nonparametric model checks for regression. Ann. Statist. 25 , 613-641 (1997) · Zbl 0926.62035
[16] Stute, W., Wang, J.-L.: The strong law under random censorship. Ann. Statist. 21 , 1591-1607 (1993) · Zbl 0785.60020
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