In this thesis, we try to provide a broad econometric analysis of a class of risk measures, distortion risk measures (DRM). With carefully selected functional form, the Value-at-Risk (VaR) and Tail-VaR (TVaR) are special cases of DRMs. Besides, the DRM also admits interpretation in the sense of non-expected utility type of preferences. We first provide a unified statistical framework for the nonparametric estimators of the DRMs in a univariate case. The asymptotic properties of both the DRMs and their sensitivities with respect to the parameters representing risk aversion and/or pessimism are derived. Moreover, the relationships between the VaR and TVaR are also investigated in detail, which, we hope, can shed new lights on the way passing one risk measure to another. Then, the analysis of DRMs are extended to a multi-dimensional framework, where the DRM is computed for a portfolio consisting of many primitive assets. Analogous to the mean-variance frontier analysis, we study the efficient portfolio frontier when both objective and constraint are replaced by the DRMs. We call this the DRM-DRM framework. Under a nonparametric setting, we propose three asymptotic test statistics for evaluating the efficiency of a given portfolio. Finally, we discuss the criteria used for evaluating models used to forecast the VaRs. More precisely, we propose a criterion which takes into account the loss levels beyond the VaRs.

Functional data analysis (FDA) is an active field of statistics, in which the primary subjects in the study are curves. My dissertation consists of two innovative applications of functional data analysis in biology. The data that motivated the research broadened the scope of FDA and demanded new methodology. I develop new nonparametric methods to make various estimations, and I focus on developing large sample theories for the proposed estimators. The first project is motivated from a colon carcinogenesis study, the goal of which is to study the function of a protein (p27) in colon cancer development. In this study, a number of colonic crypts (units) were sampled from each rat (subject) at random locations along the colon, and then repeated measurements on the protein expression level were made on each cell (subunit) within the selected crypts. In this problem, measurements within each crypt can be viewed as a function, since the measurements can be indexed by the cell locations. The functions from the same subject are spatially correlated along the colon, and my goal is to estimate this correlation function using nonparametric methods. We use this data set as an motivation and propose a kernel estimator of the correlation function in a more general framework. We develop a pointwise asymptotic normal distribution for the proposed estimator when the number of subjects is fixed and the number of units within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the unit locations. Simulation studies are also provided to illustrate the numerical performance of the proposed method. My second project is on a lipoprotein profile data, where the goal is to use lipoprotein profile curves to predict the cholesterol level in human blood. Again, motivated by the data, we consider a more general problem: the functional linear models (Ramsay and Silverman, 1997) with functional predictor and scalar response. There is literature developing different methods for this model; however, there is little theory to support the methods. Therefore, we focus more on the theoretical properties of this model. There are other contemporary theoretical work on methods based on Principal Component Regression. Our work is different in the sense that we base our method on roughness penalty approach and consider a more realistic scenario that the functional predictor is observed only on discrete points. To reduce the difficulty of the theoretical derivations, we restrict the functions with a periodic boundary condition and develop an asymptotic convergence rate for this problem in Chapter III. A more general result based on splines is a future research topic that I give some discussion in Chapter IV.

Conventional Cobb-Douglas and Transcendental Logarithmic production functions widely used in Stochastic Production Frontier Estimation and Inefficiency Analysis have merits and deficiencies. The Cobb-Douglas function imposes monotonicity and concavity constraints required by microeconomic theory. However it is inflexible and implies undesired assumptions as well. The Trans-log function is very flexible and does not imply undesired assumptions, yet it is very hard to impose both monotonicity and concavity constraints. The first essay introduced a class of stochastic production frontier estimation models that impose monotonicity and concavity constraints and suggested models that are very flexible. Researchers can use arbitrary order of polynomial functions or any function of independent variables within the suggested frameworks. Also shown was that adopting suggested models could greatly increase predictive accuracy through simulations. In the second essay we generalized the suggested models with the Inefficiency Analysis technique. In the last essay we extended the models developed in the previous two essays with regression spline and let the data decide how flexible or complicated the model should be. We showed the improvement of deterministic frontier estimation this extension could bring through simulations, as well. Works in this dissertation reduced the gap between conventional structural models and nonparametric models in stochastic frontier estimation field. This dissertation offered applied researchers Stochastic Production Frontier models that are more accurate and flexible than previous ones. It also preserves constraints of economic theory.

Nonparametric Bayesian models have been researched extensively in the past 10 years following the work of Escobar and West (1995) on sampling schemes for Dirichlet processes. The infinite mixture representation of the Dirichlet process makes it useful for clustering problems where the number of clusters is unknown. We develop nonparametric Bayesian models for two different clustering problems, namely functional and graphical clustering. We propose a nonparametric Bayes wavelet model for clustering of functional or longitudinal data. The wavelet modelling is aimed at the resolution of global and local features during clustering. The model also allows the elicitation of prior belief about the regularity of the functions and has the ability to adapt to a wide range of functional regularity. Posterior inference is carried out by Gibbs sampling with conjugate priors for fast computation. We use simulated as well as real datasets to illustrate the suitability of the approach over other alternatives. The functional clustering model is extended to analyze splice microarray data. New microarray technologies probe consecutive segments along genes to observe alternative splicing (AS) mechanisms that produce multiple proteins from a single gene. Clues regarding the number of splice forms can be obtained by clustering the functional expression profiles from different tissues. The analysis was carried out on the Rosetta dataset (Johnson et al., 2003) to obtain a splice variant by tissue distribution for all the 10,000 genes. We were able to identify a number of splice forms that appear to be unique to cancer. We propose a Bayesian model for partitioning graphs depicting dependencies in a collection of objects. After suitable transformations and modelling techniques, the problem of graph cutting can be approached by nonparametric Bayes clustering. We draw motivation from a recent work (Dhillon, 2001) showing the equivalence of kernel k-means clustering and certain graph cutting algorithms. It is shown that loss functions similar to the kernel k-means naturally arise in this model, and the minimization of associated posterior risk comprises an effective graph cutting strategy. We present here results from the analysis of two microarray datasets, namely the melanoma dataset (Bittner et al., 2000) and the sarcoma dataset (Nykter et al., 2006).

Thesis (Ph. D.)--University of California, San Diego, 2004. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2003. / Title from first page of PDF file. Document formatted into pages; contains xiii, 102 p.: ill (some col.). Includes abstract and vita. Advisor: Ayres G.D'Costa, College of Education. Includes bibliographical references (p. 97-102).

published_or_final_version / abstract / Business / Master / Master of Philosophy

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Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [121]-127).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Economics." Includes bibliographical references (p. 103-110).