Advances in modern technology have facilitated the collection of high-dimensional functional and low dimensional longitudinal data. For these data, it is often of interest to describe the key signals of the data (mean functions, covariance functions, derivative functions, etc.). Functional data analysis (FDA) and longitudinal data analysis (LDA) techniques have played a central role in the analysis of these data.The primary goal of this dissertation is to provide some novel statistical inference methods for FDA and LDA.In Chapter 1, we describe the structure (design, notations, etc.) of functional data and describe the spline smoothing technique asa tool to analysis these data. Longitudinal data analysis with missing not at random response is also discussed.In Chapter 2, a polynomial spline estimator is proposed for the mean function of dense functional data together with a simultaneous confidence band which is asymptotically correct. In addition, the spline estimator and its accompanying confidence band enjoy semiparametric efficiency in the sense that they are asymptotically the same as if all random trajectories are observed entirely and without errors. The confidence band is also extended to the difference of mean functions of two populations of functional data. Simulation experiments provide strong evidence that corroborates the asymptotic theory while computing is efficient. The confidence band procedure is illustrated by analyzing the near infrared spectroscopy data.A nonparametric estimation of the covariance function for densefunctional data using tensor product B-splines is considered in Chapter 3. We develop both local and global asymptotic distributions for the proposed estimator, and show that our estimator is as efficient as an ``oracle'' estimator. Monte Carlo simulation experiments and two real data examples are also provided to illustrate theproposed method in this chapter.In Chapter 4, we develop a new procedure to construct simultaneous confidence bands for derivatives of mean curves in FDA. The technique involves polynomial splines that provide an approximation to the derivatives of the mean functions, the covariance functions and the associated eigenfunctions. The confidence band procedure is illustrated through numerical simulation studies and a real life example.In Chapter 5, we consider data generated from a longitudinal study with potentially non random missing data. For these data, a joint model for the missing data process and the outcome process, is found to beat best weakly identifiable. Due to this identifiability concerns, tests concerning the parameters of interest may not be able to use conventional theories and it may not be clear how to assess statistical significance. We extend the literature by developing a testing procedure that can be used to evaluate hypotheses under non and weakly identifiable semiparametric models. We derive the limiting distribution of this statistic and propose theoretically justified resampling approaches to approximate its asymptotic distribution. The methodology's practical utility is illustrated in simulations and an analysis of quality-of-life outcomes from a longitudinal study on breast cancer.
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