The thesis focuses on methodological research and associated statistical theories originated from the functional data analysis perspective, and consists of three chapters. In the first chapter, we propose novel optimal designs for longitudinal data for the common situation where the resources for longitudinal data collection are limited, by determining the optimal locations in time where measurements should be taken. As for all optimal designs, some prior information is needed to implement the proposed optimal designs. We demonstrate that this prior information may come from a pilot longitudinal study that has irregularly measured and noisy measurements, where for each subject one has available a small random number of repeated measurements that are randomly located on the domain. A second possibility of interest is that a pilot study consists of densely measured functional data and one intends to take only a few measurements at strategically placed locations in the domain for the future collection of similar data. We construct optimal designs by targeting two criteria:(a) Optimal designs to recover the unknown underlying smooth random trajectory for each subject from a few optimally placed measurements such that squared prediction errors are minimized;(b) Optimal designs that minimize prediction errors for functional linear regression with functional or longitudinal predictors and scalar responses, again from a few optimally placed measurements. The proposed optimal designs address the need for sparse data collection when planning longitudinal studies, by taking advantage of the close connections between longitudinal and functional data analysis. We demonstrate in simulations that the proposed designs perform considerably better than randomly chosen design points and include a motivating data example from the Baltimore longitudinal study of aging. The proposed designs are shown to have an asymptotic optimality property. In the second chapter, we connect the optimal design method with model selection in multivariate regression framework. Model selection in linear regression models with scalar response has been a popular topic because of its important role in a vast range of applications, including high dimensional data samples with sample size n smaller than the number of predictors p. Classic model selection methods such as Lasso and state of the art methods in machine learning for the most part rely on the assumption of independent/uncorrelated covariates and sparseness of the regression coefficient vector, where most regression coefficients are assumed to be zero. However, highly correlated covariates are commonly encountered in real applications and the sparsity assumption is often hard to verify. In this work, we propose a novel methodology for model selection using stringing and the perspective of functional data analysis. Instead of sparsity and uncorrelatedness of the predictors, the stringing approach has been established under the alternative assumption that observed predictor vectors are drawn from a square integrable and smooth stochastic process with subsequent random scrambling of the order of the coordinates where the functions are sampled. We make use of recent results for optimal designs in functional regression models to develop the high-dimensional predictor selection with stringing (HIPS) approach with optimal designs, which provides covariate selection. The HIPS approach consists of three steps, namely(1) recovering the assumed underlying order of the observed high-dimensional covariates, which are assumed to be the scrambled coordinates of an underlying smooth process (the stringing step); (2) selecting the optimal design points/covariates for the recovered functional trajectories, which will then provide the set of selected covariates (the optimal design step); (3) using the selected predictors to fit the linear model. We compare the prediction performance as well as the ability of capturing the correct covariates under different scenarios of HIPS and the relaxed Lasso in simulation studies and real data applications. In the last chapter, we develop a novel method under the context of residual demography, which is a recent concept that has proved to be a useful tool to gain insights about the age distributions of wild populations, especially insects. We develop an operator equation that permits the derivation of functionals of the age distribution in wild populations, such as mean age, within the framework of residual demography. Our method combines information from an observed captive cohort, which consists of subjects that are sampled from the wild with unknown ages and then raised in the laboratory until death, and from a reference cohort that consists of subjects raised in the laboratory since birth of the same population. Targeting functionals such as the mean of the wild age distribution has the advantage of avoiding strong assumptions such as stationarity and stability of the population that one would need when targeting the entire survival distribution in thewild. Our main result characterizes the existence of a solution of the operator equation that yields the functional of interest. The proposed method also enjoys straightforward and easy implementation. A data example is included illustrating an application, whereone aims to attain the mean age of mosquitoes in the wild, based on seasonal captive cohorts from Greece and a simulated reference cohort, separately for various summer and fall months.