Nonparametric and Semiparametric Approaches to Functional Data Modeling
We propose original nonparametric and semiparametric approaches to model the relationship between a pair of variables (X, Y ), where X ∈ L2 (T ) is a square-integrable random function over a compact interval T, and Y ∈ R. Modeling of functional data greatly extends the nonparameteric approaches that have been widely used in the multivariate setting for both classifications and regressions. However, such approaches can be problematic due to the infinite-dimensional nature of functional data and the so-called curse-of-dimensionality. Fortunately, functional data often lie in a low-dimensional subspace. Therefore, one can project the data onto a subspace of dimension J, e.g., the first J principal components, where J is a tuning parameter, and the model performance is sensitive to the choice of J. Our work develops methods controlling the cut-off basis J with respect to the finite sample size n for functional data, covering both classifications and scalar-on-function regressions. A semiparametric Bayes classifier is developed using the copula structure to model dependency between the J projected scores. Furthermore, for functional local linear models, a class of multidimensional data-adaptive ridge penalties is built, and an algorithm of empirically estimating the first-order derivatives is developed. The methods prove to have strong prediction performance and dimension reduction strength when compared to other popular approaches through comprehensive simulation scenarios and real-data examples. We also point out the methods’ effective bandwidth size control, indicating their strength in finite-sample variance reduction. The estimators’ asymptotic performances are also discussed when J → ∞ as n → ∞.
Asymptotic Theory; Functional Data; Local Linear Regression; Nonparametric Statistics
Hooker, Giles J.; Matteson, David
Ph. D., Statistics
Doctor of Philosophy
dissertation or thesis