
A penalized likelihood ratio test for generalized partial functional linear models is proposed. A roughness penalty is used to control the model complexity via a smoothing parameter. A new type of inner product is defined. Using this inner product, a Bahadur representation for both functional and scalar penalized estimators is developed based on the reproducing kernel Hilbert space. The proposed approach allows for detecting the significant effects of the functional and scalar covariates on the scalar outcome, either simultaneously or separately. It is shown that the scalar estimators are asymptotically independent of the estimator of the functional part. The null limit distribution of the proposed test statistic is shown to be normal and approximately chi-square. The restriction that the scalar covariates can only linearly associate with the functional process is imposed and then the decay rates of the corresponding coefficients are determined. Simulation studies are presented to investigate the finite sample performance of the model. The simulated data are generated from the partial functional linear model and the partial functional logistic regression model. An application of the model is made to determine effects of PM2.5, the functional part formed with daily concentration measurements from April 1 to August 31, 2000, in the presence of scalar factors. The response variable is the nonaccidental mortality rate across different cities in the United States measured by the log-transformed total mortality rate in the following month, September 2000, among individuals of age 65 and older. The data set is obtained from the National Mortality, Morbidity, and Air Pollution Study. Significant effects of PM2.5, proportion of the population with at least a high school diploma, proportion of the population with at least a university diploma, and proportion of the population below the poverty line are found.
Generalized linear models (logistic models), Functional data analysis, Reliability and life testing, hypothesis testing, Bahadur representation, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Applications of statistics to environmental and related topics, penalized likelihood ratio test, reproducing kernel Hilbert space, Applications of statistics to biology and medical sciences; meta analysis
Generalized linear models (logistic models), Functional data analysis, Reliability and life testing, hypothesis testing, Bahadur representation, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Applications of statistics to environmental and related topics, penalized likelihood ratio test, reproducing kernel Hilbert space, Applications of statistics to biology and medical sciences; meta analysis
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