×
Kim, Young-Ju

Nonparametric estimation of varying-coefficient single-index models. (English) Zbl 1514.62675

J. Appl. Stat. 42, No. 2, 281-291 (2015).
Summary: The varying-coefficient single-index model has two distinguishing features: partially linear varying-coefficient functions and a single-index structure. This paper proposes a nonparametric method based on smoothing splines for estimating varying-coefficient functions and an unknown link function. Moreover, the average derivative estimation method is applied to obtain the single-index parameter estimates. For interval inference, Bayesian confidence intervals were obtained based on Bayes models for varying-coefficient functions and the link function. The performance of the proposed method is examined both through simulations and by applying it to Boston housing data.

MSC:

62-XX Statistics

Keywords:

Bayesian confidence interval; penalized likelihood; single-index; smoothing splines; varying-coefficient functions

Software:

gss
Cite Review PDF
Full Text: DOI

References:

[1] I. Ahmad, S. Leelahanon, and Q. Li, Efficient estimation of a semiparametric partially linear varying coefficient model, Ann. Stat. 33 (2005), pp. 258-283. doi: 10.1214/009053604000000931 · Zbl 1064.62043 · doi:10.1214/009053604000000931
[2] C.T. Chiang, J.A. Rice, and C.O. Wu, Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables, J. Am. Statist. Assoc. 93 (2001), pp. 961-994.
[3] P.P.B. Eggermont, R.L. Eubank, and V.N. LaRiccia, Convergence rates for smoothing spline estimators in varying coefficient models, J. Statist. Plann. Inference 140 (2010), pp. 369-381. doi: 10.1016/j.jspi.2009.06.017 · Zbl 1177.62049 · doi:10.1016/j.jspi.2009.06.017
[4] J. Fan and T. Huang, Profile likelihood inference on semiparametric varying-coefficient partially linear models, Bernoulli 11 (2005), pp. 1031-1057. doi: 10.3150/bj/1137421639 · Zbl 1098.62077 · doi:10.3150/bj/1137421639
[5] C. Gu, Smoothing Spline ANOVA models, 2nd ed., Springer, New York, 2013. · Zbl 1269.62040 · doi:10.1007/978-1-4614-5369-7
[6] C. Gu and Y.-J. Kim, Penalized likelihood regression: General formulation and efficient approximation, Can. J. Stat. 30 (2002), pp. 619-628. doi: 10.2307/3316100 · Zbl 1018.62032 · doi:10.2307/3316100
[7] W. Hardle and T.M. Stoker, Investigating smooth multiple regression by the method of average derivatives, J. Am. Statist. Assoc. 84 (1989), pp. 986-995. · Zbl 0703.62052
[8] W. Hardle, P. Hall, and H. Ichimura, Optimal smoothing in single-index models, Ann. Stat. 21 (1993), pp. 157-178. doi: 10.1214/aos/1176349020 · Zbl 0770.62049 · doi:10.1214/aos/1176349020
[9] T.J. Hastie and R. Tibshirani, Varying-coefficient models, J. R. Stat. Soc. Ser. B 55 (1993), pp. 757-796. · Zbl 0796.62060
[10] T.J. Hastie, R. Tibshirani, and J. Friedman, The elements of statistical learning, Springer, New York, 2001. · Zbl 0973.62007 · doi:10.1007/978-0-387-21606-5
[11] D.R. Hoover, J.A. Rice, C.O. Wu, and L-P. Yang, Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data, Biometricka 85 (1998), pp. 809-822. doi: 10.1093/biomet/85.4.809 · Zbl 0921.62045 · doi:10.1093/biomet/85.4.809
[12] M. Hristache, A. Juditsky, J. Polzehl, and V. Spokoiny, Structure adaptive approach for dimension reduction, Ann. Stat. 29 (2001), pp. 1537-1566. · Zbl 1043.62052
[13] T. Hu and Y. Xia, Adpative semi-varying coefficient model selection, Statist. Sin. 22 (2012), pp. 575-599. · Zbl 1511.62168
[14] Z. Huang, Efficient inferences on the varying-coefficient single-index model with empirical likelihood, Comput. Statist. Data Anal. 56 (2012), pp. 4413-4420. doi: 10.1016/j.csda.2012.03.024 · Zbl 1255.62090 · doi:10.1016/j.csda.2012.03.024
[15] Z. Huang and R. Zhang, Empirical likelihood for the varying-coefficient single-index model, Can. J. Stat. 38 (2010), pp. 434-452. doi: 10.1002/cjs.10075 · Zbl 1327.62260 · doi:10.1002/cjs.10075
[16] Y-J. Kim, A partial spline approach for semiparametric estimation of varying-coefficient partially linear models, Comput. Stat. Data Anal. 62 (2013), pp. 181-187. doi: 10.1016/j.csda.2013.01.006 · Zbl 1348.62122 · doi:10.1016/j.csda.2013.01.006
[17] Y-J. Kim and C. Gu, Smoothing spline Gaussian regression: More scalable computation via efficient approximation, J. R. Statist. Soc. Ser. B 66 (2004), pp. 337-356. doi: 10.1046/j.1369-7412.2003.05316.x · Zbl 1062.62067 · doi:10.1046/j.1369-7412.2003.05316.x
[18] F. Leitenstorfer and G. Tutz, Estimation of single-index models based on boosting techniques, Stat. Model. 11 (2011), pp. 203-217. doi: 10.1177/1471082X1001100302 · doi:10.1177/1471082X1001100302
[19] C. Leng, A simple approach for varying-coefficient model selection, J. Statist. Plann. Inference 139 (2009), pp. 2138-2146. doi: 10.1016/j.jspi.2008.10.009 · Zbl 1160.62067 · doi:10.1016/j.jspi.2008.10.009
[20] D. Nychka, Bayesian confidence intervals for smoothing splines, J. Am. Statist. Assoc. 83 (1988), pp. 1134-1143. doi: 10.1080/01621459.1988.10478711
[21] J. Polzehl and S. Sperlich, A note on structural adaptive dimension reduction, J. Statist. Comput. Simul. 79 (2009), pp. 805-818. doi: 10.1080/00949650801959699 · Zbl 1186.62084
[22] G. Wahba, Bayesian confidence interval for the cross-validated smoothing spline, J. R. Statist. Soc. Ser. B 45 (1983), pp. 133-150. · Zbl 0538.65006
[23] Q. Wang and L. Xue, Statistical inference in partially-varying-coefficient single-index model, J. Multivariate Anal. 102 (2011), pp. 1-19. doi: 10.1016/j.jmva.2010.07.005 · Zbl 1206.62113 · doi:10.1016/j.jmva.2010.07.005
[24] H. Wong, W-C. Ip, and R. Zhang, Varying-coefficient single-index model, Comput. Statist. Data Anal. 52 (2008), pp. 1458-1476. doi: 10.1016/j.csda.2007.04.008 · Zbl 1452.62296 · doi:10.1016/j.csda.2007.04.008
[25] Y. Xia, H. Tong, W. K. Li, and L. Zhn, An adaptive estimation of dimension reduction, J. R. Statist. Soc. Ser. B 64 (2002), pp. 363-410. doi: 10.1111/1467-9868.03411 · Zbl 1091.62028 · doi:10.1111/1467-9868.03411
[26] Y. Xia, W. Zhang, and H. Tong, Efficient estimation for semivarying-coefficient models, Biometrika 91 (2004), pp. 661-681. doi: 10.1093/biomet/91.3.661 · Zbl 1108.62019 · doi:10.1093/biomet/91.3.661
[27] Y. Yu and D. Ruppert, Penalized spline estimation for partially linear single-index models, J. Am. Statist. Assoc. 97 (2002), pp. 1042-1054. doi: 10.1198/016214502388618861 · Zbl 1045.62035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.