Estimation of a semiparametric transformation model. (English) Zbl 1133.62029

Summary: This paper proposes consistent estimators for transformation parameters in semiparametric models. The problem is to find the optimal transformation into the space of models with a predetermined regression structure like additive or multiplicative separability. We give results for the estimation of the transformation when the rest of the model is estimated non- or semi-parametrically and fulfills some consistency conditions.
We propose two methods for the estimation of the transformation parameter: maximizing a profile likelihood function or minimizing the mean squared distance from independence. First the problem of identification of such models is discussed. We then state asymptotic results for a general class of nonparametric estimators. Finally, we give some particular examples of nonparametric estimators of transformed separable models. The small sample performance is studied in several simulations.


62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
62F12 Asymptotic properties of parametric estimators
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[1] Amemiya, T. and Powell, J. L. (1981). A comparison of the Box-Cox maximum likelihood estimator and the nonlinear two-stage least squares estimator. J. Econometrics 17 351-381. · Zbl 0488.62096
[2] Bickel, P. J. and Doksum, K. (1981). An analysis of transformations revisited. J. Amer. Statist. Assoc. 76 296-311. JSTOR: · Zbl 0464.62058
[3] Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations. J. Roy. Statist. Soc. Ser. B 26 211-252. JSTOR: · Zbl 0156.40104
[4] Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80 580-619. JSTOR: · Zbl 0594.62044
[5] Carroll, R. J. and Ruppert, D. (1984). Power transformation when fitting theoretical models to data. J. Amer. Statist. Assoc. 79 321-328. JSTOR:
[6] Carroll, R. J. and Ruppert, D. (1988). Transformation and Weighting in Regression . Chapman and Hall, New York. · Zbl 0666.62062
[7] Chen, X., Linton, O. B. and Van Keilegom, I. (2003). Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71 1591-1608. JSTOR: · Zbl 1154.62325
[8] Cheng, S. C., Wei, L. J. and Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika 82 835-845. JSTOR: · Zbl 0861.62071
[9] Cheng, K. F. and Wu, J. W. (1994). Adjusted least squares estimates for the scaled regression coefficients with censored data. J. Amer. Statist. Assoc. 89 1483-1491. JSTOR: · Zbl 0810.62063
[10] Doksum, K. (1987). An extension of partial likelihood methods for proportional hazard models to general transformation models. Ann. Statist. 15 325-345. · Zbl 0639.62026
[11] Ekeland, I., Heckman, J. J. and Nesheim, L. (2004). Identification and estimation of Hedonic Models. J. Political Economy 112 S60-S109.
[12] Hall, P. and Horowitz, J. L. (1996). Bootstrap critical values for tests based on generalized-method-of-moments estimators. Econometrica 64 891-916. JSTOR: · Zbl 0854.62045
[13] Hengartner, N. W. and Sperlich, S. (2005). Rate optimal estimation with the integration method in the presence of many covariates. J. Multivariate Anal. 95 246-272. · Zbl 1070.62021
[14] Horowitz, J. (1996). Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64 103-137. JSTOR: · Zbl 0861.62029
[15] Horowitz, J. (2001). Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69 499-513. JSTOR: · Zbl 0999.62032
[16] Ibragimov, I. A. and Hasminskii, R. Z. (1980). On nonparametric estimation of regression. Soviet Math. Dokl. 21 810-814. · Zbl 0516.62061
[17] Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika 36 149-176. JSTOR: · Zbl 0033.07204
[18] Koul, H. L. (2001). Weighted Empirical Processes in Regression and Autoregression Models . Springer, New York.
[19] Lewbel, A. and Linton, O. (2007). Nonparametric matching and efficient estimators of homothetically separable functions. Econometrica 75 1209-1227. · Zbl 1134.91548
[20] Linton, O. B., Chen, R., Wang, N. and Härdle, W. (1997). An analysis of transformations for additive nonparametric regression. J. Amer. Statist. Assoc. 92 1512-1521. JSTOR: · Zbl 0912.62048
[21] Linton, O. and Mammen, E. (2005). Estimating semiparametric ARCH(\infty ) models by kernel smoothing. Econometrica 73 771-836. JSTOR: · Zbl 1153.91798
[22] Linton, O. B. and Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression using marginal integration. Biometrika 82 93-100. JSTOR: · Zbl 0823.62036
[23] Mammen, E., Linton, O. B. and Nielsen, J. P. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443-1490. · Zbl 0986.62028
[24] Mammen, E. and Park, B. U. (2005). Bandwidth selection for smooth backfitting in additive models. Ann. Statist. 33 1260-1294. · Zbl 1072.62025
[25] Nielsen, J. P., Linton, O. B. and Bickel, P. J. (1998). On a semiparametric survival model with flexible covariate effect. Ann. Statist. 26 215-241. · Zbl 0953.62107
[26] Nielsen, J. P. and Sperlich, S. (2005). Smooth backfitting in practice. J. Roy. Statist. Soc. Ser. B 61 43-61. · Zbl 1060.62048
[27] Robinson, P. M. (1991). Best nonlinear three-stage least squares estimation of certain econometric models. Econometrica 59 755-786. JSTOR: · Zbl 0729.62106
[28] Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768-1802. · Zbl 0768.62015
[29] Sperlich, S. (2005). On nonparametric estimation with constructed variables and generated regressors. Preprint, Univ. Carlos III de Madrid, Spain.
[30] Sperlich, S., Linton, O. B. and Härdle, W. (1999). Integration and backfitting methods in additive models: Finite sample properties and comparison. Test 8 419-458. · Zbl 0938.62045
[31] Sperlich, S., Linton, O. B. and Van Keilegom, I. (2007). A computational note on estimation of a semiparametric transformation model. Preprint, Georg-August Univ. Göttingen, Germany. · Zbl 1133.62029
[32] Sperlich, S., Tjøstheim, D. and Yang, L. (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18 197-251. · Zbl 1109.62310
[33] Stone, C. J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 1348-1360. · Zbl 0451.62033
[34] Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040-1053. · Zbl 0511.62048
[35] Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 592-606. · Zbl 0603.62050
[36] Tjøstheim, D. and Auestad, B. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398-1409. · Zbl 0813.62036
[37] van den Berg, G. J. (2001). Duration models: Specification, identification and multiple durations. In The Handbook of Econometrics V (J. J. Heckman and E. Leamer, eds.) 3381-3460. North-Holland, Amsterdam.
[38] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[39] Van Keilegom, I. and Veraverbeke, N. (2002). Density and hazard estimation in censored regression models. Bernoulli 8 607-625. · Zbl 1007.62029
[40] Wei, L. J. (1992). The accelerated failure time model: A useful alternative to the Cox regression model in survival analysis. Statistics in Medicine 11 1871-1879.
[41] Zellner, A. and Revankar, N. S. (1969). Generalized production functions. Rev. Economic Studies 36 241-250. · Zbl 0176.50103
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