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Estimation and variable selection for generalized additive partial linear models. (English) Zbl 1227.62053
Summary: We study generalized additive partial linear models, proposing the use of polynomial spline smoothing for estimation of nonparametric functions, and deriving quasi-likelihood based estimators for the linear parameters. We establish asymptotic normality for the estimators of the parametric components. The procedure avoids solving large systems of equations as in kernel-based procedures and thus results in gains in computational simplicity. We further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi-likelihood, which is shown to have an asymptotic oracle property. Monte Carlo simulations and an empirical example are presented for illustration.

MSC:
62J12 Generalized linear models (logistic models)
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
65D10 Numerical smoothing, curve fitting
65C05 Monte Carlo methods
Software:
SemiPar
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[1] Breiman, L. (1996). Heuristics of instability and stabilization in model selection. Ann. Statist. 24 2350-2383. · Zbl 0867.62055 · doi:10.1214/aos/1032181158
[2] Buja, A., Hastie, T. and Tibshirani, R. (1989). Linear smoothers and additive models (with discussion). Ann. Statist. 17 453-555. · Zbl 0689.62029 · doi:10.1214/aos/1176347115
[3] Carroll, R. J., Fan, J. Q., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477-489. · Zbl 0890.62053 · doi:10.2307/2965697
[4] Carroll, R. J., Maity, A., Mammen, E. and Yu, K. (2009). Nonparametric additive regression for repeatedly measured data. Biometrika 96 383-398. · Zbl 1163.62028 · doi:10.1093/biomet/asp015
[5] Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions. Numer. Math. 31 377-403. · Zbl 0377.65007 · doi:10.1007/BF01404567 · eudml:132586
[6] de Boor, C. (2001). A Practical Guide to Splines , revised ed. Applied Mathematical Sciences 27 . Springer, New York. · Zbl 0987.65015
[7] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348-1360. · Zbl 1073.62547 · doi:10.1198/016214501753382273
[8] Fan, J. and Li, R. (2006). Statistical challenges with high dimensionality: Feature selection in knowledge discovery. In International Congress of Mathematicians. Vol. III 595-622. Eur. Math. Soc., Zürich. · Zbl 1117.62137
[9] Frank, I. E. and Friedman, J. H. (1993). A statistical view of some chemometrics regression tools (with discussion). Technometrics 35 109-148. · Zbl 0775.62288 · doi:10.2307/1269656
[10] Härdle, W., Huet, S., Mammen, E. and Sperlich, S. (2004). Bootstrap inference in semiparametric generalized additive models. Econometric Theory 20 265-300. · Zbl 1072.62034 · doi:10.1017/S026646660420202X
[11] Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Monographs on Statistics and Applied Probability 43 . Chapman and Hall, London. · Zbl 0747.62061
[12] Huang, J. (1998). Functional ANOVA models for generalized regression. J. Multivariate Anal. 67 49-71. · Zbl 0949.62028 · doi:10.1006/jmva.1998.1753
[13] Huang, J. (1999). Efficient estimation of the partially linear additive Cox model. Ann. Statist. 27 1536-1563. · Zbl 0977.62035 · doi:10.1214/aos/1017939141
[14] Hunter, D. and Li, R. (2005). Variable selection using MM algorithms. Ann. Statist. 33 1617-1642. · Zbl 1078.62028 · doi:10.1214/009053605000000200
[15] Li, R. and Liang, H. (2008). Variable selection in semiparametric regression modeling. Ann. Statist. 36 261-286. · Zbl 1132.62027 · doi:10.1214/009053607000000604 · euclid:aos/1201877301
[16] Li, Y. and Ruppert, D. (2008). On the asymptotics of penalized splines. Biometrika 95 415-436. · Zbl 1437.62540 · doi:10.1093/biomet/asn010
[17] Lin, X. and Carroll, R. J. (2006). Semiparametric estimation in general repeated measures problems. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 69-88. · Zbl 1141.62026 · doi:10.1111/j.1467-9868.2005.00533.x
[18] Linton, O. and Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93-101. · Zbl 0823.62036 · doi:10.1093/biomet/82.1.93
[19] Marx, B. D. and Eilers, P. H. C. (1998). Direct generalized additive modeling with penalized likelihood. Comput. Statist. Data Anal. 28 193-209. · Zbl 1042.62580 · doi:10.1016/S0167-9473(98)00033-4
[20] McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models , 2nd ed. Monographs on Statistics and Applied Probability 37 . Chapman and Hall, London. · Zbl 0744.62098
[21] Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models. J. Roy. Statist. Soc. Ser. A 135 370-384.
[22] Ruppert, D., Wand, M. and Carroll, R. (2003). Semiparametric Regression . Cambridge Univ. Press, Cambridge. · Zbl 1038.62042
[23] Severini, T. A. and Staniswalis, J. G. (1994). Quasi-likelihood estimation in semiparametric models. J. Amer. Statist. Assoc. 89 501-511. · Zbl 0798.62046 · doi:10.2307/2290852
[24] Smith, J. W., Everhart, J. E., Dickson, W. C., Knowler, W. C. and Johannes, R. S. (1988). Using the ADAP learning algorithm to forecast the onset of diabetes mellitus. In Proc. Annu. Symp. Comput. Appl. Med. Care 261-265. IEEE Computer Society Press, Washington, DC.
[25] Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705. · Zbl 0605.62065 · doi:10.1214/aos/1176349548
[26] Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606. · Zbl 0603.62050 · doi:10.1214/aos/1176349940
[27] Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Statist. 22 118-184. · Zbl 0827.62038 · doi:10.1214/aos/1176325361
[28] Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. Ser. B Stat. Methodol. 58 267-288. · Zbl 0850.62538
[29] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics . Springer, New York. · Zbl 0862.60002
[30] Wood, S. N. (2004). Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Assoc. 99 673-686. · Zbl 1117.62445 · doi:10.1198/016214504000000980 · masetto.asa.catchword.org
[31] Wood, S. N. (2006). Generalized Additive Models . Chapman & Hall/CRC Press, Boca Raton, FL. · Zbl 1087.62082
[32] Xue, L. and Yang, L. (2006). Additive coefficient modeling via polynomial spline. Statist. Sinica 16 1423-1446. · Zbl 1109.62030
[33] Yu, K. and Lee, Y. K. (2010). Efficient semiparametric estimation in generalized partially linear additive models. J. Korean Statist. Soc. 39 299-304. · Zbl 1294.62090 · doi:10.1016/j.jkss.2010.02.001
[34] Yu, K., Park, B. U. and Mammen, E. (2008). Smooth backfitting in generalized additive models. Ann. Statist. 36 228-260. · Zbl 1132.62028 · doi:10.1214/009053607000000596 · euclid:aos/1201877300
[35] Zou, H. (2006). The adaptive Lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418-1429. · Zbl 1171.62326 · doi:10.1198/016214506000000735
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