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Nonparametric \(n^{-1/2}\)-consistent estimation for the general transformation models. (English) Zbl 0894.62049
Summary: We propose simple estimators for the transformation function \(\Lambda\) and the distribution function \(F\) of the error for the model \[ \Lambda(Y)=\alpha+X\beta+\varepsilon. \] 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 J. L. Horowitz [Econometrica 64, No. 1, 103-137 (1996; Zbl 0861.62029)].
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.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI
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[25] CHICAGO, ILLINOIS 60637 1700 MAIN STREET E-MAIL: jmy@gsbjmy.uchicago.edu SANTA MONICA, CALIFORNIA 90401 E-MAIL: naihua duan@rand.org
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