Direct estimation of low-dimensional components in additive models
Jianqing Fan, Wolfgang Härdle, and Enno Mammen
Source: Ann. Statist. Volume 26, Number 3
(1998), 943-971.
Abstract
Additive regression models have turned out to be a useful statistical tool in analyses of high-dimensional data sets. Recently, an estimator of additive components has been introduced by Linton and Nielsen which is based on marginal integration. The explicit definition of this estimator makes possible a fast computation and allows an asymptotic distribution theory. In this paper an asymptotic treatment of this estimate is offered for several models. A modification of this procedure is introduced. We consider weighted marginal integration for local linear fits and we show that this estimate has the following advantages.
(i) With an appropriate choice of the weight function, the additive components can be efficiently estimated: An additive component can be estimated with the same asymptotic bias and variance as if the other components were known.
(ii) Application of local linear fits reduces the design related bias.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1024691083
Mathematical Reviews number (MathSciNet): MR1635422
Digital Object Identifier: doi:10.1214/aos/1024691083
Zentralblatt MATH identifier: 01359546
References
BERNDT, E. R. 1991. The Practice of Econometrics: Classic and Contemporary. Addison-Wesley, Reading, MA. Z.
BHATTACHARy A, P. K. and ZHAO, P.-L. 1997. Semiparametric inference in a partial linear model. Ann. Statist. 25 244 262. Z. Z
Mathematical Reviews (MathSciNet): MR98f:62081
Zentralblatt MATH: 0869.62050
Digital Object Identifier: doi:10.1214/aos/1034276628
Project Euclid: euclid.aos/1034276628
BUJA, A., HASTIE, T. J. and TIBSHIRANI, R. J. 1989. Linear smoothers and additive models with. discussion. Ann. Statist. 17 453 510. Z.
Mathematical Reviews (MathSciNet): MR994249
Zentralblatt MATH: 0689.62029
Digital Object Identifier: doi:10.1214/aos/1176347115
Project Euclid: euclid.aos/1176347115
CARROLL, R. J., FAN, J., GIJBELS, I. and WAND, M. P. 1997. Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477 489. Z.
Mathematical Reviews (MathSciNet): MR98f:62215
Zentralblatt MATH: 0890.62053
Digital Object Identifier: doi:10.2307/2965697
JSTOR: links.jstor.org
CHEN, R., HARDLE, W., LINTON, O. and SEVERANCE-LOSSIN, E. 1996. Estimation and variable ¨ selection in additive nonparametric regression models. In Statistical Theory and Z. Computational Aspects of Smoothing W. Hardle and M. Schimek, eds.. physika, ¨ Heidelberg. Z.
FAN, J. 1993. Local linear regression smoothers and their minimax efficiency. Ann. Statist. 21 196 216. Z.
Mathematical Reviews (MathSciNet): MR94c:62060
Zentralblatt MATH: 0773.62029
Digital Object Identifier: doi:10.1214/aos/1176349022
Project Euclid: euclid.aos/1176349022
FAN, J. 1997. Comments on ``Poly nomial splines and their tensor product in the extended linear models'' by C. J. Stone, M. H. Hansen, C. Kooperberg and Y. U. Troung. Ann. Statist. 25 1425 1432. Z.
FAN, J. and GIBELS, I. 1992. Variable bandwidth and local linear regression smoothers. Ann. Statist. 20 2008 2036. Z.
Mathematical Reviews (MathSciNet): MR1193323
Zentralblatt MATH: 0765.62040
Digital Object Identifier: doi:10.1214/aos/1176348900
Project Euclid: euclid.aos/1176348900
FAN, J. and GIJBELS, I. 1996. Local Poly nomial Modeling and Its Applications. Chapman and Hall, London. Z.
Mathematical Reviews (MathSciNet): MR1383587
Zentralblatt MATH: 0873.62037
FRANZ, W. 1991. Arbeitsokonomik. Springer, Berlin. ¨ Z.
GASSER, T. and MULLER, H.-G. 1979. Kernel estimation of regression functions. Smoothing ¨ Techniques for Curve Estimation. Lecture Notes in Math. 757 23 68. Springer, New York. Z.
Mathematical Reviews (MathSciNet): MR81k:62052
Digital Object Identifier: doi:10.1007/BFb0098489
HARDLE, W. and MAMMEN, E. 1993. Testing parametric versus nonparametric regression. Ann. ¨ Statist. 21 1926 1947.
HARDLE, W., MAMMEN, E. and MULLER, M. 1995. Testing parametric versus semiparametric ¨ ¨ modelling in generalized linear models. Technical Report. Z.
HARDLE, W. and TSy BAKOV, A. B. 1995. Additive nonparametric regression on principal compo¨ nents, J. Nonparametr. Statist. 5 157 184. Z.
HASTIE, T. J. and TIBSHIRANI, R. J. 1990. Generalized Additive Models. Chapman and Hall, London. Z.
Mathematical Reviews (MathSciNet): MR92e:62117
HENGARTNER, N. W. 1996. Rate optimal estimation of additive regression via the integration method in the presence of many covariates. Unpublished manuscript. Z.
LINTON, O. B. 1997. Efficient estimation of additive nonparametric regression models. Biometrika 84 469 473. Z.
Mathematical Reviews (MathSciNet): MR98k:62062
Zentralblatt MATH: 0882.62038
Digital Object Identifier: doi:10.1093/biomet/84.2.469
JSTOR: links.jstor.org
LINTON, O. B., MAMMEN, E. and NIELSEN, J. P. 1997. The existence and asy mptotic properties of a backfitting projection algorithm under weak conditions. Preprint. Z.
LINTON, O. B. and NIELSEN, J. P. 1995. A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93 101. Z.
Mathematical Reviews (MathSciNet): MR96f:62065
Zentralblatt MATH: 0823.62036
Digital Object Identifier: doi:10.1093/biomet/82.1.93
JSTOR: links.jstor.org
MACK, Y. P. and SILVERMAN, B. W. 1982. Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 61 405 415. Z.
Mathematical Reviews (MathSciNet): MR84h:62072
Zentralblatt MATH: 0495.62046
Digital Object Identifier: doi:10.1007/BF00539840
OPSOMER, J. D. 1997. On the existence and asy mptotic properties of backfitting estimators. Preprint. Z.
OPSOMER, J. D. and RUPPERT, D. 1997. Fitting a bivariate additive model by local poly nomial regression. Ann. Statist. 25 186 211. Z.
Mathematical Reviews (MathSciNet): MR1429922
Zentralblatt MATH: 0869.62026
Digital Object Identifier: doi:10.1214/aos/1034276626
Project Euclid: euclid.aos/1034276626
RUPPERT, D. and WAND, M. P. 1994. Multivariate weighted least squares regression. Ann. Statist. 22 1346 1370. Z.
Mathematical Reviews (MathSciNet): MR1311979
Zentralblatt MATH: 0821.62020
Digital Object Identifier: doi:10.1214/aos/1176325632
Project Euclid: euclid.aos/1176325632
SPECKMAN, P. 1988. Kernel smoothing in partial linear models. J. Roy. Statist. Soc. Ser. B 50 413 436. Z.
STONE, C. J. 1983. Optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives. In Recent Advances in Statistics: Papers Presented Z in Honor of Herman Chernoff's Sixtieth Birthday M. H. Rizvi, J. S. Rustagi and D.. Siegmund, eds.. Academic Press, New York. Z.
Mathematical Reviews (MathSciNet): MR85m:62086
Zentralblatt MATH: 0591.62031
STONE, C. J. 1985. Additive regression and other nonparametric models. Ann. Statist. 13 685 705. Z.
Mathematical Reviews (MathSciNet): MR87i:62111
Zentralblatt MATH: 0605.62065
Digital Object Identifier: doi:10.1214/aos/1176349548
Project Euclid: euclid.aos/1176349548
STONE, C. J. 1986. The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 592 606. Z. TJøSTHEIM, D. and AUESTAD, B. H. 1994. Nonparametric identification of nonlinear time series: projections. J. Amer. Statist. Assoc. 89 1398 1409. Z.
TREIMAN, D. J. 1978. Probleme der Begriffsbildung und Operationalisierung in der international vergleichenden Mobilitatsforschung. In Sozialstrukturanalysen mit Umfrage¨ Z. daten F. U. Pappi, ed.. Athenaum, Kronberg im Taunus. ¨
CHAPEL HILL, NORTH CAROLINA 27599-3260 HUMBOLDT-UNIVERSITAT ZU BERLIN ¨ SPANDAUER STRASSE 1 10178 BERLIN GERMANY
