Simultaneous Confidence Bands For Sparse Functional Data And High Dimensional Data

In this dissertation, we focus on how to construct data driven simultaneous confi-dence bands for unknown nonparametric functions in two different models, for sparse functional data and high dimensional data, respectively.One is the varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time independent covariates with coefficients as functions of time dependent covariates. Based on spline smoothing, we propose data driven simultaneous confidence bands for the coefficient functions with asymptotically correct confidence level. Such confidence bands are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypothe-ses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p-values on the effects of smoking, pre-infection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment.The other is the popular semiparametric single-index model for high dimensional data. Over the last twenty-five years, various.(?)n-consistent estimators have been devised for the coefficient vector in the single-index model. We prove under general assumptions that kernel estimator of the link function by univariate regression on the single-index variable is oracally efficient, namely, the estimator with the true single-index coefficient vector is asymptotically indistinguishable from that with any (?)n-consistent coefficient vector estimator. As a mathematical by product of the oracle efficiency, simultaneous confidence band is constructed for the link function based on the oracally efficient kernel estimator. Simulation experiments support the asymptotic theories. The proposed simultaneous confidence band is applied to analyze and test hypothesis about the Boston housing data.