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

Efficient nonparametric regression when the support is bounded

We consider nonparametric regression models with error distributions that have bounded support and are (possibly) nonregular in the sense that sufficient mass is concentrated in the neighbourhood of its endpoint(s). Models of this type arise naturally, e.g., in the analysis of auctions and records, production frontiers and in image analysis. Moreover, in extreme value analysis of environmental data with covariates one is frequently confronted with potentially nonregular models of the aforementioned type.

It has been shown that, using the nonregularity, the regression function can be estimated much more precisely than in regular models. However, the fundamental structure of such nonregular infinite-dimensional regression models is not yet well understood. In this project, we will first establish asymptotic approximations of these nonregular statistical experiments by simpler ones that allow to focus on essential features of nonregular models. Optimal estimators in these limit experiments then yield optimal estimators for the unknown regression function.

These estimators will depend both on the (non)regularity of the error distribution and the degree of smoothness of the regression function, which are usually not known in practical applications. Hence to facilitate the considerable gain of efficiency offered by the model structure, we will develop data-driven methods to simultaneously adapt both to the regularity and the smoothness. Besides estimation procedures, we shall also be interested in nonparametric inference on the regression function, especially adaptive testing and adaptive confidence bands. Moreover, tests for assumptions on the error distribution and on the value of the regularity parameter will be constructed.


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