Semi-parametric transformation boundary regression models
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Semi-parametric transformation boundary regression models Natalie Neumeyer1 · Leonie Selk1 · Charles Tillier1 Received: 4 December 2018 / Revised: 13 June 2019 © The Institute of Statistical Mathematics, Tokyo 2019
Abstract In the context of nonparametric regression models with one-sided errors, we consider parametric transformations of the response variable in order to obtain independence between the errors and the covariates. In view of estimating the transformation parameter, we use a minimum distance approach and show the uniform consistency of the estimator under mild conditions. The boundary curve, i.e., the regression function, is estimated applying a smoothed version of a local constant approximation for which we also prove the uniform consistency. We deal with both cases of random covariates and deterministic (fixed) design points. To highlight the applicability of the procedures and to demonstrate their performance, the small sample behavior is investigated in a simulation study using the so-called Yeo–Johnson transformations. Keywords Box–Cox transformations · Frontier estimation · Minimum distance estimation · Local constant approximation · Boundary models · Nonparametric regression · Yeo–Johnson transformations
Financial support by the DFG (Research Unit FOR 1735 Structural Inference in Statistics: Adaptation and Efficiency) is gratefully acknowledged. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10463019-00731-5) contains supplementary material, which is available to authorized users.
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Charles Tillier [email protected] Natalie Neumeyer [email protected] Leonie Selk [email protected]
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Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
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N. Neumeyer et al.
1 Introduction Before fitting a regression model, it is very common in applications to transform the response variable. The aim of the transformation is to gain efficiency in the statistical inference, for instance, by reducing skewness or inducing a specific structure of the model, e.g., linearity of the regression function or homoscedasticity. In practice, often a parametric class of transformations is considered from which an ‘optimal’ one should be selected data dependently (with a specific purpose in mind). A classical example is the class of Box–Cox power transformations introduced for linear models by Box and Cox (1964). There is a vast literature on parametric transformation models in the context of mean regression, and we refer to the monograph by Carroll and Ruppert (1988). Powell (1991) introduced Box–Cox transformations in the context of linear quantile regression; see also Mu and He (2007) who considered transformations to obtain a linear quantile regression function. Horowitz (2009) reviewed estimation in transformation models with parametric regression in the cases where either the transformation or the error distribution or both are modeled nonparametrically. Linton et al. (2008) suggested parametric
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