Open Access
November 2019 Estimation of fully nonparametric transformation models
Benjamin Colling, Ingrid Van Keilegom
Bernoulli 25(4B): 3762-3795 (November 2019). DOI: 10.3150/19-BEJ1110


Consider the following nonparametric transformation model $\Lambda (Y)=m(X)+\varepsilon $, where $X$ is a $d$-dimensional covariate, $Y$ is a continuous univariate dependent variable and $\varepsilon $ is an error term with zero mean and which is independent of $X$. We assume that the unknown transformation $\Lambda $ is strictly increasing and that $m$ is an unknown regression function. Our goal is to develop two new nonparametric estimators of the transformation $\Lambda $, the first one based on the least squares loss and the second one based on the least absolute deviation loss, and to compare their performance with that of the estimators developed by Chiappori, Komunjer and Kristensen (J. Econometrics 188 (2015) 22–39). Our proposed estimators are based on an estimator of the conditional distribution of $U$ given $X$, where $U$ is an appropriate transformation of $Y$ that is uniformly distributed. The main motivation for working with $U$ instead of $Y$ is that, in transformation models, the response $Y$ is often skewed with very long tails, and so kernel smoothing based on $Y$ does not work well. Hence, we expect to obtain better estimators if we pre-transform $Y$ before applying kernel smoothing. We establish the asymptotic normality of the two proposed estimators. We also carry out a simulation study to illustrate the performance of our estimators, to compare these new estimators with the ones of Chiappori, Komunjer and Kristensen (J. Econometrics 188 (2015) 22–39) and to see under which model conditions which estimators behave the best.


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Benjamin Colling. Ingrid Van Keilegom. "Estimation of fully nonparametric transformation models." Bernoulli 25 (4B) 3762 - 3795, November 2019.


Received: 1 August 2018; Revised: 1 January 2019; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110155
MathSciNet: MR4010972
Digital Object Identifier: 10.3150/19-BEJ1110

Keywords: asymptotic properties , Identification , kernel smoothing , Nonparametric regression , nonparametric transformation

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4B • November 2019
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