Nonparametric Detection of Arbitrary Changes to Distributions and Methods of Regularization of Piecewise Constant Functional Data

Nonparametric statistical methods can refer a wide variety of techniques. In this dissertation, we focus on two problems in statistics which are common applications of nonparametric statistics. The main body of the dissertation focuses on distribution-free process control for detection of arbitrary changes to the distribution of an underlying random variable. A secondary problem, also part of the broad umbrella of nonparametric statistics, is the proper approximation of a function. Statistical process control minimizes disruptions to a properly controlled process and quickly terminates out of control processes. Although rarely satisfied in practice, strict distributional assumptions are often needed to monitor these processes. Previous models have often exclusively focused on monitoring changes in the mean or variance of the underlying process. The proposed model establishes a monitoring method requiring few distributional assumptions while monitoring all changes in the underlying distribution generating the data. No assumptions on the form of the in-control distribution are made other than independence within and between observed samples. Windowing is employed to reduce computational complexity of the algorithm as well as ensure fast detection of changes. Results indicate quicker detection of large jumps than in many previously established methods. It is now common to analyze large quantities of data generated by sensors over time. Traditional analysis techniques do not incorporate the inherent functional structure often present in this type of data. The second focus of this dissertation is the development of a analysis method for functional data where the range of the function has a discrete, ordinal structure. Use is made of spline based methods using a piecewise constant function approximation. After a large amount of data reduction is achieved, generalized linear mixed model methodology is employed in order to model the data.