Estimating and testing of functional data with restrictions

The objective of this dissertation is to develop a suitable statistical methodology for functional data analysis. Modern advanced technology allows researchers to collect samples as functional which means the ideal unit of samples is a curve. We consider each functional observation as the resulting of a digitized recoding or a realization from a stochastic process. Traditional statistical methodologies often fail to be applied to this functional data set due to the high dimensionality. Functional hypothesis testing is the main focus of my dissertation. We suggested a testing procedure to determine the significance of two curves with order restriction. This work was motivated by a case study involving high-dimensional and high-frequency tidal volume traces from the New York State Psychiatric Institute at Columbia University. The overall goal of the study was to create a model of the clinical panic attack, as it occurs in panic disorder (PD), in normal human subjects. We proposed a new dimension reduction technique by non-negative basis matrix factorization (NBMF) and adapted a one-degree of freedom test in the context of multivariate analysis. This is important because other dimension techniques, such as principle component analysis (PCA), cannot be applied in this context due to the order restriction. Another area that we investigated was the estimation of functions with constrained restrictions such as convexification and/or monotonicity, together with the development of computationally efficient algorithms to solve the constrained least square problem. This study, too, has potential for applications in various fields. For example, in economics the cost function of a perfectly competitive firm must be increasing and convex, and the utility function of an economic agent must be increasing and concave. We propose an estimation method for a monotone convex function that consists of two sequential shape modification stages: (i) monotone regression via solving a constrained least square problem and (ii) convexification of the monotone regression estimate via solving an associated constrained uniform approximation problem.