×

Polynomial spline estimation for partial functional linear regression models. (English) Zbl 1347.65036

Summary: Because of its orthogonality, interpretability and best representation, functional principal component analysis approach has been extensively used to estimate the slope function in the functional linear model. However, as a very popular smooth technique in nonparametric/semiparametric regression, polynomial spline method has received little attention in the functional data case. In this paper, we propose the polynomial spline method to estimate a partial functional linear model. Some asymptotic results are established, including asymptotic normality for the parameter vector and the global rate of convergence for the slope function. Finally, we evaluate the performance of our estimation method by some simulation studies.

MSC:

62-08 Computational methods for problems pertaining to statistics
62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference

Software:

fda (R)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716-723 · Zbl 0314.62039 · doi:10.1109/TAC.1974.1100705
[2] Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76:1102-1110 · Zbl 1090.62036 · doi:10.1016/j.spl.2005.12.007
[3] Aneiros-Pérez G, Vieu P (2008) Nonparametric time series prediction: a semi-functional partial linear modeling. J Multivar Anal 99:834-857 · Zbl 1133.62075 · doi:10.1016/j.jmva.2007.04.010
[4] Baíllo A, Grané A (2009) Local linear regression for functional predictor and scalar response. J Multivar Anal 100:102-111 · Zbl 1151.62028 · doi:10.1016/j.jmva.2008.03.008
[5] Burba F, Ferraty F, Vieu P (2009) K-nearest neighbour method in functional nonparametric regression. J Nonparametric Stat 21:453-469 · Zbl 1161.62017 · doi:10.1080/10485250802668909
[6] Cai TT, Hall P (2006) Prediction in functional linear regression. Annals Stat 34:2159-2179 · Zbl 1106.62036 · doi:10.1214/009053606000000830
[7] Cai Z, Fan J, Yao Q (2000) Functional-coefficient regression model for nonlinear time series. J Am Stat Assoc 95:941-956 · Zbl 0996.62078 · doi:10.1080/01621459.2000.10474284
[8] Cardot H, Ferraty F, Sarda P (1999) Functional linear model. Stat Prob Lett 45:11-22 · Zbl 0962.62081 · doi:10.1016/S0167-7152(99)00036-X
[9] Cardot H, Ferraty F, Sarda P (2003) Spline estimators for the functional linear model. Stat Sin 13:571-591 · Zbl 1050.62041
[10] Cardot H, Sarda P (2008) Varying-coefficient functional linear regression models. Commun Stat Theory Methods 37:3186-3203 · Zbl 1292.62053 · doi:10.1080/03610920802105176
[11] Chen H (1991) Polynomial splines and nonparametric regression. J Nonparametric Stat 1:143-156 · Zbl 1263.62050 · doi:10.1080/10485259108832516
[12] Crambes C, Kneip A, Sarda P (2009) Smoothing splines estimators for functional linear regression. Annals Stat 37:35-72 · Zbl 1169.62027 · doi:10.1214/07-AOS563
[13] de Boor C (2001) A practical guide to splines. Springer, New York · Zbl 0987.65015
[14] DeVore RA, Lorentz GG (1993) Constructive approximation. Springer, Berlin · Zbl 0797.41016 · doi:10.1007/978-3-662-02888-9
[15] Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New York · Zbl 1119.62046
[16] Huang JZ (2003a) Asymptotics for polynomial spline regression under weak conditions. Stat Probab Lett 65:207-216 · Zbl 1048.62043 · doi:10.1016/j.spl.2003.09.003
[17] Huang JZ (2003b) Local asymptotics for polynomial spline regression. Annals Stat 31:1600-1635 · Zbl 1042.62035 · doi:10.1214/aos/1065705120
[18] Huang JZ, Wu CO, Zhou L (2004a) Polynomial spline estimation and inference for varying coefficient model with longitudinal data. Stat Sin 14:763-788 · Zbl 1073.62036
[19] Huang JZ, Shen H (2004b) Functional coefficient regression models for non-linear time series: a polynomial spline approach. Scand J Stat 31:515-534 · Zbl 1062.62184 · doi:10.1111/j.1467-9469.2004.00404.x
[20] Hall P, Horowitz J (2007) Methodology and convergence rates for functional linear regression. Annals Stat 35:70-91 · Zbl 1114.62048 · doi:10.1214/009053606000000957
[21] Li Y, Hsing T (2007) On rates of convergence in functional linear regression. J Multivar Anal 98:1782-1804 · Zbl 1130.62035 · doi:10.1016/j.jmva.2006.10.004
[22] Lian H (2011) Functional partial linear model. J Nonparametric Stat 23:115-128 · Zbl 1359.62157 · doi:10.1080/10485252.2010.500385
[23] Newey WK (1997) Convergence rates and asymptotic normality for series estimators. J Econom 79:147-168 · Zbl 0873.62049 · doi:10.1016/S0304-4076(97)00011-0
[24] Ramsay J, Dalzell C (1991) Some tools for functional data analysis. J R Stat Soc Ser B 53:539-572 · Zbl 0800.62314
[25] Ramsay J, Silverman B (1997) Functional data analysis. Springer, New York · Zbl 0882.62002 · doi:10.1007/978-1-4757-7107-7
[26] Ramsay J, Silverman B (2005) Functional data analysis, 2nd edn. Springer, New York · Zbl 1079.62006
[27] Ramsay J, Hooker G, Graves S (2009) Functional data analysis with R and Matlab. Springer, New York · Zbl 1179.62006 · doi:10.1007/978-0-387-98185-7
[28] Rice JA, Silverman BW (1991) Estimating the mean and covariance structure nonparametrically when the data are curves. J R Stat Soc Ser B 53:233-243 · Zbl 0800.62214
[29] Shin H (2009) Partial functional linear regression. J Stat Plan Inference 139:3405-3418 · Zbl 1168.62358 · doi:10.1016/j.jspi.2009.03.001
[30] Stone CJ (1994) The use of polynomial splines and their tensor products in multivariate function estimation (with discussion). Annals Stat 22:118-171 · Zbl 0827.62038 · doi:10.1214/aos/1176325361
[31] Stone CJ, Hansen M, Kooperberg C, Truong YK (1997) Polynomial splines and their tensor products in extended linear modeling (with discussion). Annals Stat 25:1371-1470 · Zbl 0924.62036 · doi:10.1214/aos/1031594728
[32] Schwarz G (1978) Estimating the dimension of a model. Annals Stat 6:461-464 · Zbl 0379.62005 · doi:10.1214/aos/1176344136
[33] Zhang D, Lin X, Sowers M (2007) Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome. Biometrics 63:351-362 · Zbl 1147.62391 · doi:10.1111/j.1541-0420.2006.00713.x
[34] Zhou S, Shen X, Wolfe DA (1998) Local asymptotics for regression splines and confidence regions. Annals Stat 26:1760-1782 · Zbl 0929.62052 · doi:10.1214/aos/1024691356
[35] Zhou J, Chen M (2012) Spline estimators for semi-functional linear model. Stat Probab Lett 82:505-513 · Zbl 1237.62051 · doi:10.1016/j.spl.2011.11.027
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.