Mixed models, posterior means and penalized least squares

In recent years there has been increased research activity in the area of Func- tional Data Analysis. Methodology from finite dimensional multivariate analysis has been extended to the functional data setting giving birth to Functional ANOVA, Functional Principal Components Analysis, etc. In particular, some studies have pro- posed inferential techniques for various functional models that have connections to well known areas such as mixed-effects models or spline smoothing. The methodol- ogy used in these cases is computationally intensive since it involves the estimation of coefficients in linear models, adaptive selection of smoothing parameters, estimation of variances components, etc. This dissertation proposes a wide-ranging modeling framework that includes many functional linear models as special cases. Three widely used tools are con- sidered: mixed-effects models, penalized least squares, and Bayesian prediction. We show that, in certain important cases, the same numerical answer is obtained for these seemingly different techniques. In addition, under certain assumptions, an applica- tion of a Kalman filter algorithm is shown to improve the order of computations, by two orders of magnitude, for point and interval estimates (with n being the sample size). A functional data analysis setting is used to exemplify our results.