The Annals of Statistics
- Ann. Statist.
- Volume 36, Number 2 (2008), 686-718.
Estimation of a semiparametric transformation model
Oliver Linton, Stefan Sperlich, and Ingrid Van Keilegom
Full-text: Open access
Abstract
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.
Article information
Source
Ann. Statist., Volume 36, Number 2 (2008), 686-718.
Dates
First available in Project Euclid: 13 March 2008
Permanent link to this document
https://projecteuclid.org/euclid.aos/1205420516
Digital Object Identifier
doi:10.1214/009053607000000848
Mathematical Reviews number (MathSciNet)
MR2396812
Zentralblatt MATH identifier
1133.62029
Subjects
Primary: 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 62G05: Estimation 62G08: Nonparametric regression 62G20: Asymptotic properties
Keywords
Additive models generalized structured models profile likelihood semiparametric models separability transformation models
Citation
Linton, Oliver; Sperlich, Stefan; Van Keilegom, Ingrid. Estimation of a semiparametric transformation model. Ann. Statist. 36 (2008), no. 2, 686--718. doi:10.1214/009053607000000848. https://projecteuclid.org/euclid.aos/1205420516
References
- 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.Mathematical Reviews (MathSciNet): MR659799
Digital Object Identifier: doi:10.1016/0304-4076(81)90007-5 - Bickel, P. J. and Doksum, K. (1981). An analysis of transformations revisited. J. Amer. Statist. Assoc. 76 296–311.Mathematical Reviews (MathSciNet): MR624332
Digital Object Identifier: doi:10.2307/2287831
JSTOR: links.jstor.org
Zentralblatt MATH: 0464.62058 - Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations. J. Roy. Statist. Soc. Ser. B 26 211–252.
- Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80 580–619.Mathematical Reviews (MathSciNet): MR803258
Digital Object Identifier: doi:10.2307/2288473
JSTOR: links.jstor.org
Zentralblatt MATH: 0594.62044 - Carroll, R. J. and Ruppert, D. (1984). Power transformation when fitting theoretical models to data. J. Amer. Statist. Assoc. 79 321–328.Mathematical Reviews (MathSciNet): MR755088
Digital Object Identifier: doi:10.2307/2288271
JSTOR: links.jstor.org - Carroll, R. J. and Ruppert, D. (1988). Transformation and Weighting in Regression. Chapman and Hall, New York.
- 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.Mathematical Reviews (MathSciNet): MR2000259
Digital Object Identifier: doi:10.1111/1468-0262.00461
JSTOR: links.jstor.org
Zentralblatt MATH: 1154.62325 - Cheng, S. C., Wei, L. J. and Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika 82 835–845.Mathematical Reviews (MathSciNet): MR1380818
Zentralblatt MATH: 0861.62071
Digital Object Identifier: doi:10.1093/biomet/82.4.835
JSTOR: links.jstor.org - 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.Mathematical Reviews (MathSciNet): MR1310237
Digital Object Identifier: doi:10.2307/2291010
JSTOR: links.jstor.org
Zentralblatt MATH: 0810.62063 - Doksum, K. (1987). An extension of partial likelihood methods for proportional hazard models to general transformation models. Ann. Statist. 15 325–345.Mathematical Reviews (MathSciNet): MR885740
Digital Object Identifier: doi:10.1214/aos/1176350269
Project Euclid: euclid.aos/1176350269
Zentralblatt MATH: 0639.62026 - Ekeland, I., Heckman, J. J. and Nesheim, L. (2004). Identification and estimation of Hedonic Models. J. Political Economy 112 S60–S109.
- Hall, P. and Horowitz, J. L. (1996). Bootstrap critical values for tests based on generalized-method-of-moments estimators. Econometrica 64 891–916.Mathematical Reviews (MathSciNet): MR1399222
Digital Object Identifier: doi:10.2307/2171849
JSTOR: links.jstor.org
Zentralblatt MATH: 0854.62045 - 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.Mathematical Reviews (MathSciNet): MR2170397
Digital Object Identifier: doi:10.1016/j.jmva.2004.09.010
Zentralblatt MATH: 1070.62021 - Horowitz, J. (1996). Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64 103–137.Mathematical Reviews (MathSciNet): MR1366143
Digital Object Identifier: doi:10.2307/2171926
JSTOR: links.jstor.org
Zentralblatt MATH: 0861.62029 - Horowitz, J. (2001). Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69 499–513.Mathematical Reviews (MathSciNet): MR1819761
Digital Object Identifier: doi:10.1111/1468-0262.00200
JSTOR: links.jstor.org
Zentralblatt MATH: 0999.62032 - Ibragimov, I. A. and Hasminskii, R. Z. (1980). On nonparametric estimation of regression. Soviet Math. Dokl. 21 810–814.
- Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika 36 149–176.
- Koul, H. L. (2001). Weighted Empirical Processes in Regression and Autoregression Models. Springer, New York.
- Lewbel, A. and Linton, O. (2007). Nonparametric matching and efficient estimators of homothetically separable functions. Econometrica 75 1209–1227.Mathematical Reviews (MathSciNet): MR2333499
Digital Object Identifier: doi:10.1111/j.1468-0262.2007.00787.x
Zentralblatt MATH: 1134.91548 - 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.Mathematical Reviews (MathSciNet): MR1615261
Digital Object Identifier: doi:10.2307/2965422
JSTOR: links.jstor.org
Zentralblatt MATH: 0912.62048 - Linton, O. and Mammen, E. (2005). Estimating semiparametric ARCH(∞) models by kernel smoothing. Econometrica 73 771–836.Mathematical Reviews (MathSciNet): MR2135143
Digital Object Identifier: doi:10.1111/j.1468-0262.2005.00596.x
JSTOR: links.jstor.org
Zentralblatt MATH: 1153.91798 - Linton, O. B. and Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression using marginal integration. Biometrika 82 93–100.Mathematical Reviews (MathSciNet): MR1332841
Zentralblatt MATH: 0823.62036
Digital Object Identifier: doi:10.1093/biomet/82.1.93
JSTOR: links.jstor.org - 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.
- Mammen, E. and Park, B. U. (2005). Bandwidth selection for smooth backfitting in additive models. Ann. Statist. 33 1260–1294.Mathematical Reviews (MathSciNet): MR2195635
Digital Object Identifier: doi:10.1214/009053605000000101
Project Euclid: euclid.aos/1120224102
Zentralblatt MATH: 1072.62025 - 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.Mathematical Reviews (MathSciNet): MR1611784
Digital Object Identifier: doi:10.1214/aos/1030563983
Project Euclid: euclid.aos/1030563983
Zentralblatt MATH: 0953.62107 - Nielsen, J. P. and Sperlich, S. (2005). Smooth backfitting in practice. J. Roy. Statist. Soc. Ser. B 61 43–61.Mathematical Reviews (MathSciNet): MR2136638
Digital Object Identifier: doi:10.1111/j.1467-9868.2005.00487.x
Zentralblatt MATH: 1060.62048 - Robinson, P. M. (1991). Best nonlinear three-stage least squares estimation of certain econometric models. Econometrica 59 755–786.Mathematical Reviews (MathSciNet): MR1106511
Digital Object Identifier: doi:10.2307/2938227
JSTOR: links.jstor.org
Zentralblatt MATH: 0729.62106 - Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768–1802.Mathematical Reviews (MathSciNet): MR1193312
Digital Object Identifier: doi:10.1214/aos/1176348889
Project Euclid: euclid.aos/1176348889
Zentralblatt MATH: 0768.62015 - Sperlich, S. (2005). On nonparametric estimation with constructed variables and generated regressors. Preprint, Univ. Carlos III de Madrid, Spain.
- 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.
- 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.
- Sperlich, S., Tjøstheim, D. and Yang, L. (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18 197–251.
- Stone, C. J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 1348–1360.Mathematical Reviews (MathSciNet): MR594650
Digital Object Identifier: doi:10.1214/aos/1176345206
Project Euclid: euclid.aos/1176345206
Zentralblatt MATH: 0451.62033 - Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053.Mathematical Reviews (MathSciNet): MR673642
Digital Object Identifier: doi:10.1214/aos/1176345969
Project Euclid: euclid.aos/1176345969
Zentralblatt MATH: 0511.62048 - Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 592–606.Mathematical Reviews (MathSciNet): MR840516
Digital Object Identifier: doi:10.1214/aos/1176349940
Project Euclid: euclid.aos/1176349940
Zentralblatt MATH: 0603.62050 - Tjøstheim, D. and Auestad, B. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398–1409.
- 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.
- Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
- Van Keilegom, I. and Veraverbeke, N. (2002). Density and hazard estimation in censored regression models. Bernoulli 8 607–625.
- 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.
- Zellner, A. and Revankar, N. S. (1969). Generalized production functions. Rev. Economic Studies 36 241–250.
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