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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.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference
SemiPar
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##### References:
 [1] BAHADUR, R. R. 1966. A note on quantiles in large samples. Ann. Math. Statist. 37 577 580. Z. · Zbl 0147.18805 [2] BICKEL, P. J. and DOKSUM, K. A. 1981. An analysis of transformations revisited. J. Amer. Statist. Assoc. 76 296 311. Z. Z. JSTOR: · Zbl 0464.62058 [3] BOX, G. E. P. and COX, D. R. 1964. An analysis of transformations with discussion. J. Roy. Statist. Soc. Ser B 26 211 252. Z. JSTOR: · Zbl 0156.40104 [4] BOX, G. E. P. and COX, D. R. 1982. Comment on “An analysis of transformations revisited,” by P. J. Bickel and K. A. Doksum. J. Amer. Statist. Assoc. 77 209 210. Z. JSTOR: · Zbl 0504.62058 [5] BREIMAN, L. and FRIEDMAN, J. H. 1985. Estimating optimal transformations for multiple regression and correlation, J. Amer. Statist. Assoc. 80 580 598. Z. JSTOR: · Zbl 0594.62044 [6] CARROLL, R. J. and RUPPERT, D. 1988. Transformation and Weighting in Regression. Chapman and Hall, London. Z. · Zbl 0666.62062 [7] CHAUDHURI, P., DOKSUM, K. and SAMAROV, A. 1994. Nonparametric estimation of global functionals based on quantile regression. Unpublished manuscript. Z. · Zbl 0885.62042 [8] DOKSUM, K. 1987. An extension of partial likelihood methods for proportional hazard models to general transformation models. Ann. Statist. 15 325 345. Z. · Zbl 0639.62026 [9] DUAN, N. 1990. The adjoint projection pursuit regression. J. Amer. Statist. Assoc. 85 1029 1038. Z. JSTOR: · Zbl 0736.62056 [10] EFRON, B. 1982. Transformation theory: how normal is a family of distributions? Ann. Statist. 10 323 339. Z. · Zbl 0507.62008 [11] FRIEDMAN, J. H. and STUETZLE, W. 1981. Projection pursuit regression. J. Amer. Statist. Assoc. 76 817 823. Z. JSTOR: [12] HARDLE, W. and STOKER, T. 1989. Investigating smooth multiple regression by the method of äverage derivatives. J. Amer. Statist. Assoc. 84 986 995. Z. JSTOR: · Zbl 0703.62052 [13] HECKMAN, J. and SINGER, B. 1984. A method for minimizing the impact of distributional assumptions in econometric models for duration data. Econometrica 52 271 320. Z. Z. JSTOR: · Zbl 0547.62077 [14] HINKLEY, D. V. and RUNGER, G. 1984. The analysis of transformed data with discussion. J. Amer. Statist. Assoc. 79 302 320. Z. JSTOR: · Zbl 0553.62051 [15] HOROWITZ, J. L. 1996. Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64 103 137. Z. JSTOR: · Zbl 0861.62029 [16] KRUSKAL, J. B. 1965. Analy sis of factorial experiments by estimating monotone transformations of the data. J. Roy. Statist. Soc. Ser. B 27 251 263. Z. JSTOR: [17] MURPHY, S. A. 1994. Consistency in a proportional hazards model incorporating a random effect. Ann. Statist. 22 712 731. · Zbl 0827.62033 [18] MURPHY, S. A. 1995. Asy mptotic theory for the frailty model. Ann. Statist. 23 182 198. Z. · Zbl 0822.62069 [19] PAKES, A. and POLLARD, D. 1989. Simulation and the asy mptotics of optimization estimators. Econometrica 57 1027 1057. Z. JSTOR: · Zbl 0698.62031 [20] POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z. · Zbl 0544.60045 [21] SERFLING, R. 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York. Z. · Zbl 0538.62002 [22] SHEN, X. 1997. On methods of sieves and penalization. Ann. Statist. 25 2555 2591. Z. · Zbl 0895.62041 [23] WANG, N. and RUPPERT, D. 1995. Nonparametric estimation of the transformation in the transformation-both-sides regression model. J. Amer. Statist. Assoc. 90 522 534. Z. JSTOR: · Zbl 0826.62028 [24] WANG, N. and RUPPERT, D. 1996. Estimation of regression parameters in a semiparametric transformation model. J. Statist. Plann. Inference 52 331 351. · Zbl 0960.62511 [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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