Estimating and testing of functional data with restrictions

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