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.

The embrittlement of reactor pressure vessel (RPV) steels due to neutron irradiation restricts the operating lifetime of nuclear reactors. The reference temperature 𝑇0, obtained from fracture mechanics testing using the Master Curve concept, is a good indicator of the irradiation resistance of a material. The measurement of the shift in 𝑇0 after neutron irradiation, which accompanies the embrittlement of the material, using the Master Curve concept, enables the assessment of the reactor materials. In the context of worldwide life time extensions of nuclear power plants, the limited availability of neutron irradiated materials (surveillance materials) is a challenge. Testing of miniaturized 0.16T C(T) specimens manufactured from already tested standard Charpy-sized specimens helps to solve the material shortage problem. In this work, four different reactor pressure vessel steels with different compositions were investigated in the unirradiated and in the neutron-irradiated condition. A total number of 189 mini-C(T) samples were fabricated and tested. An important component of this study is the transferability of fracture mechanics data from mini-C(T) to standard Charpy-sized specimen. Our results demonstrate good agreement of the reference temperatures from the mini-C(T) specimens with those from standard Charpy-sized specimens. RPV steels containing higher Cu and P contents exhibit a higher increase in 𝑇0 after irradiation. The fracture surfaces were investigated using SEM in order to record the location of the fracture initiators. The fracture modes were also determined. A large number of test results formed the basis for a censoring probability function, which was used to optimally select the testing temperature in Master Curve testing. The effect of the slow stable crack growth censoring criteria from ASTM E1921 on the determination of 𝑇0 was analysed and found to have a minor effect. Our results demonstrate the validity of mini-C(T) specimen testing and confirm the role of the impurity elements Cu and P in neutron embrittlement. We anticipate further research linking microstructure to the fracture properties of materials before and after neutron irradiation and the optimization of Master Curve testing using the results from our statistical analysis.

The master's thesis is focused on the posibility of laboratory testing of shotcrete. The important point of the thesis is properties comparison of shotcrete made in laboratory with vibrating press and concrete with the same recipe made by spraying machine. The influence of shotcrete recipe, type and dosage of accelerating ingredient is also examined.