Inference for Generalized partial functional linear regression

In this study, we examine inferences (in particular, hypothesis tests) for generalized partial functional linear models. A Bahadur representation for both functional and scalar estimators is developed based on the reproducing kernel Hilbert space. We establish the asymptotic independence between the scalar estimators and the estimator of the functional part. A penalized likelihood ratio test is proposed to detect the significant effects of the functional and scalar covariates on the scalar outcome, either simultaneously or separately. The asymptotic normality of the test statistic is established under the null hypothesis. Simulation studies provide numerical support for the asymptotic properties. Lastly, data on air pollution are used to demonstrate our method.

[1]  Li Hsu,et al.  Hypothesis testing in functional linear models , 2017, Biometrics.

[2]  Fang Yao,et al.  Partially functional linear regression in high dimensions , 2016 .

[3]  P. Yu,et al.  A test of linearity in partial functional linear regression , 2016 .

[4]  Raymond J Carroll,et al.  Generalized Functional Linear Models With Semiparametric Single-Index Interactions , 2010, Journal of the American Statistical Association.

[5]  Peter F. de Jong,et al.  A central limit theorem for generalized quadratic forms , 1987 .

[6]  Guang Cheng,et al.  Joint asymptotics for semi-nonparametric regression models with partially linear structure , 2013, 1311.2628.

[7]  Sara van de Geer,et al.  Penalized quasi-likelihood estimation in partial linear models , 1997 .

[8]  James O. Ramsay,et al.  Applied Functional Data Analysis: Methods and Case Studies , 2002 .

[9]  Guang Cheng,et al.  Nonparametric inference in generalized functional linear models , 2014, 1405.6655.

[10]  Nadine Hilgert,et al.  Minimax adaptive tests for the Functional Linear model , 2012, 1206.1194.

[11]  Chong Gu.  Smoothing Spline Anova Models , 2002 .

[12]  T. Tony Cai,et al.  Minimax and Adaptive Prediction for Functional Linear Regression , 2012 .

[13]  Raymond K. W. Wong,et al.  Partially Linear Functional Additive Models for Multivariate Functional Data , 2018, Journal of the American Statistical Association.

[14]  Jin-Ting Zhang,et al.  Statistical inferences for functional data , 2007, 0708.2207.

[15]  P. Hall,et al.  Properties of principal component methods for functional and longitudinal data analysis , 2006, math/0608022.

[16]  M. Yuan,et al.  A Reproducing Kernel Hilbert Space Approach to Functional Linear Regression , 2010, 1211.2607.

[17]  Ana-Maria Staicu,et al.  Classical testing in functional linear models , 2016, Journal of nonparametric statistics.

[18]  Xiao Wang,et al.  Penalized likelihood functional regression , 2014 .

[19]  Hyejin Shin,et al.  On prediction rate in partial functional linear regression , 2012, J. Multivar. Anal..