Chapter 1 proposes multifractal analysis to measure inhomogeneity of regularity of 1H-NMR spectrum using wavelet-based multifractal tools. The geometric summaries of multifractal spectrum are informative summaries, and as such employed to discriminate 1H-NMR spectra associated with different treatments. The novel summaries are based on the descriptors, originally introduced by Shi et al. (2005). The methodology is applied to evaluate the effect of sulfur amino acids. With only six univariate summaries, two statistical models capture the effect of sulfur amino acids.;Chapter 2 provides essential materials for understanding engineering background of a nano-particle fabrication process. In particular, certain physics of the engineering process are described and process outcomes are quantified. Several noise factors contributing to process uncertainties are identified. Preliminary analyses based on data obtained from computer simulations of physical experiment show the potential of further statistical modeling research opportunities.;Chapter 3 develops a two-part model for observations from nano-particle fabrication experiments. Since there are certain combinations of process variables resulting to unproductive process outcomes, a logistic model is used to characterize such a process behavior. For the cases with productive outcomes a normal regression serves the second part of the model. Because the data are obtained from computer experiments, random-effects are included in both logistics and normal regression models to describe the potential spatial correlation among data. The likelihood function for this two-part model is complicated and thus the maximum likelihood estimation is intractable. This chapter researches approximation techniques based on Taylor series extension to simplify the likelihood. An algorithm is developed to find estimates for maximizing the approximated likelihood.;Chapter 4 presents a method to decide the sample size under multi-layer system. The multi-layer is a series of layers, which become smaller and smaller. Our focus is to decide the sample size in each layer. The sample size decision has several objectives, and the most important purpose is the sample size should be enough to give a right direction to the next layer. Specifically, the bottom layer, which is the smallest neighborhood around the optimum, should meet the tolerance requirement. Other objectives considered are budget limit and model improvement. Performing the hypothesis test of whether the next layer includes the optimum gives the required sample size. We demonstrate an illustrative example to evaluate the proposed methodology.
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