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December 1997 Nonparametric $n\sp {-1/2}$-consistent estimation for the general transformation models
Jianming Ye, Naihua Duan
Ann. Statist. 25(6): 2682-2717 (December 1997). DOI: 10.1214/aos/1030741091


We propose simple estimators for the transformation function $\Lambda$ and the distribution function F of the error for the model $$\Lambda (Y) = \alpha + \mathbf{X} \mathbf{\beta} + \varepsilon.$$ It is proved that these estimators are consistent and can achieve the unusual $n^{-1/2}$ rate of convergence on any finite interval under some regularity conditions. We show that our estimators are more attractive than another class of estimators proposed by Horowitz. Interesting decompositions of the estimators are obtained. The estimator of F is independent of the unknown transformation function $\Lambda$, and the variance of the estimator for $\Lambda$ depends on $\Lambda$ only through the density function of X. Through simulations, we find that the procedure is not sensitive to the choice of bandwidth, and the computation load is very modest. In almost all cases simulated, our procedure works substantially better than median nonparametric regression.


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Jianming Ye. Naihua Duan. "Nonparametric $n\sp {-1/2}$-consistent estimation for the general transformation models." Ann. Statist. 25 (6) 2682 - 2717, December 1997.


Published: December 1997
First available in Project Euclid: 30 August 2002

zbMATH: 0894.62049
MathSciNet: MR1604440
Digital Object Identifier: 10.1214/aos/1030741091

Primary: 62G07
Secondary: 62G02

Keywords: General transformation models , mean integrated square error , median nonparametric regression , prediction interval , shifted median estimator , Uniform convergence

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 6 • December 1997
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