Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Research Unit 1735

Efficient nonparametric regression when the support is bounded

We consider nonparametric regression models where the support of the errors is bounded from at least one side, so that the regression function can be determined from the boundary of the support of the observations. Models of this type arise naturally, e.g., in the analysis of auctions, when dealing with bid and ask data from stock exchange, in the domain of production frontiers or image analysis. Particular emphasis is put on the nonregular case in which the error distribution concentrates sufficient mass in the neighbourhood of its endpoint(s).
In contrast to classical nonparametric mean regression estimators, which are based on local averages of the response random variables, boundary regression estimators depend on extreme observations. In cases where both estimating approaches are directly comparable (e.g., for symmetric error distributions with bounded support), the latter converge at a much faster rate if the error distribution is nonregular.
Unlike the structure of regular mean regression models, which has been thoroughly studied, nonparametric boundary regression experiments are less well understood, in particular if the regression function is assumed smooth, but not necessarily monotone or convex. We will approximate such experiments in which both the boundary function and the behaviour of the error distribution locally at the boundary of its support are unknown by simpler experiments. This way we will be able to construct optimal estimators and testing procedures which adapt to the unknown regularity of the boundary function and of the error distribution. Adaptive confidence bands for the regression function as well as hypotheses tests for assumptions on the error distribution and the structure of the regression function will be developed. Jointly with Project 5 (estimation of high-dimensional matrices) the new methods will be applied to estimate the realised volatility from observations of bid and ask prices.


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