We look at three distinct topics in analysis. In the first we give a direct and easy proof that the usual Newton-Leibniz rule implies the fundamental theorem of algebra that any nonconstant complex polynomial of one complex variable has a complex root. Next, we look at the Riesz representation theorem and show that the Riesz representing measure often can be given in the form of mini sums just like in the case of the usual Lebesgue measure on a cube. Lastly, we look at the idea of holomorphic domination and use it to define a class of complex Banach manifolds that is similar in nature and definition to the class of Stein manifolds.

The receiver operating characteristic (ROC) curves is a popular tool for evaluating continuous diagnostic tests. The traditional definition of ROC curves incorporates implicitly the idea of "hard" thresholding, which also results in the empirical curves being step functions. The first topic is to introduce a novel definition of soft ROC curves, which incorporates the idea of "soft" thresholding. The softness of a soft ROC curve is controlled by a regularization parameter that can be selected suitably by a cross-validation procedure. A byproduct of the soft ROC curves is that the corresponding empirical curves are smooth. The second topic is on combination of several diagnostic tests to achieve better diagnostic accuracy. We consider the optimal linear combination that maximizes the area under the receiver operating characteristic curve (AUC); the estimates of the combination's coefficients can be obtained via a non-parametric procedure. However, for estimating the AUC associated with the estimated coefficients, the apparent estimation by re-substitution is too optimistic. To adjust for the upward bias, several methods are proposed. Among them the cross-validation approach is especially advocated, and an approximated cross-validation is developed to reduce the computational cost. Furthermore, these proposed methods can be applied for variable selection to select important diagnostic tests. However, the above best-subset variable selection method is not practical when the number of diagnostic tests is large. The third topic is to further develop a LASSO-type procedure for variable selection. To solve the non-convex maximization problem in the proposed procedure, an efficient algorithm is developed based on soft ROC curves, difference convex programming, and coordinate descent algorithm.

In the dissertation, we study the statistical evaluation of treatment comparisons by evaluating the relative comparison of survival experiences between two treatment groups. We construct confidence interval and simultaneous confidence bands for the ratio and odds ratio of two survival functions through both parametric and nonparametric approaches.We first construct empirical likelihood confidence interval and simultaneous confidence bands for the odds ratio of two survival functions to address small sample efficacy and sufficiency. The empirical log-likelihood ratio is developed, and the corresponding asymptotic distribution is derived. Simulation studies show that the proposed empirical likelihood band has outperformed the normal approximation band in small sample size cases in the sense that it yields closer coverage probabilities to chosen nominal levels.Furthermore, in order to incorporate prognostic factors for the adjustment of survival functions in the comparison, we construct simultaneous confidence bands for the ratio and odds ratio of survival functions based on both the Cox model and the additive risk model. We develop simultaneous confidence bands by approximating the limiting distribution of cumulative hazard functions by zero-mean Gaussian processes whose distributions can be generated through Monte Carlo simulations. Simulation studies are conducted to evaluate the performance for proposed models. Real applications on published clinical trial data sets are also studied for further illustration purposes.In the end, the population attributable fraction function is studied to measure the impact of risk factors on disease incidence in the population. We develop semiparametric estimation of attributable fraction functions for cohort studies with potentially censored event time under the additive risk model.

In diagnostic test studies, one crucial task is to evaluate the diagnostic accuracy of a test. Currently, most studies focus on the Receiver Operating Characteristics Curve and the Area Under the Curve. On the other hand, the Youden index, widely applied in practice, is another comprehensive measurement for the performance of a diagnostic test. For a continuous-scale test classifying diseased and non-diseased groups, finding the Youden index of the test is equivalent to maximize the sum of sensitivity and specificity for all the possible values of the cut-point. This dissertation concentrates on statistical inferences for the Youden index. First, an auxiliary tool for the Youden index, called the diagnostic curve, is defined and used to evaluate the diagnostic test. Second, in the paired-design study to assess the diagnostic accuracy of two biomarkers, the difference in paired Youden indices frequently acts as an evaluation standard. We propose an exact confidence interval for the difference in paired Youden indices based on generalized pivotal quantities. A maximum likelihood estimate-based interval and a bootstrap-based interval are also included in the study. Third, for certain diseases, an intermediate level exists between diseased and non-diseased status. With such concern, we define the Youden index for three ordinal groups, propose the empirical estimate of the Youden index, study the asymptotic properties of the empirical Youden index estimate, and construct parametric and nonparametric confidence intervals for the Youden index. Finally, since covariates often affect the accuracy of a diagnostic test, therefore, we propose estimates for the Youden index with a covariate adjustment under heteroscedastic regression models for the test results. Asymptotic properties of the covariate-adjusted Youden index estimators are investigated under normal error and non-normal error assumptions.

In the first part of the thesis we present several interior point algorithms for solving certain positive definite programming problems. One of the algorithms is adapted for finding out whether there exists or not a positive definite matrix which is a real linear combination of some given symmetric matrices A1,A2, . . . ,Am. In the second part of the thesis we discuss stability of nonlinear dynamical systems. We search using algorithms described in the first part, for Lyapunov functions of a few forms. A suitable Lyapunov function implies the existence of a hyperellipsoidal attraction region for the dynamical system, thus guaranteeing stability.

In this thesis, we apply smoothed empirical likelihood method to investigate confidence intervals for the receiver operating characteristic (ROC) curve with right censoring. As a particular application of comparison of distributions from two populations, the ROC curve is constructed by the combination of cumulative distribution function and quantile function. Under mild conditions, the smoothed empirical likelihood ratio converges to chi-square distribution, which is the well-known Wilks's theorem. Furthermore, the performances of the empirical likelihood method are also illustrated by simulation studies in terms of coverage probability and average length of confidence intervals. Finally, a primary biliary cirrhosis data is used to illustrate the proposed empirical likelihood procedure.

Currently, clustering applications use classical methods to partition a set of data (or objects) in a set of meaningful sub-classes, called clusters. A cluster is therefore a collection of objects which are “similar” among them, thus can be treated collectively as one group, and are “dissimilar” to the objects belonging to other clusters. However, there are a number of problems with clustering. Among them, as mentioned in [Datta03], dealing with large number of dimensions and large number of data items can be problematic because of computational time. In this thesis, we investigate all clustering algorithms used in [Datta03] and we present a parallel solution to minimize the computational time. We apply parallel programming techniques to the statistical algorithms as a natural extension to sequential programming technique using R. The proposed parallel model has been tested on a high throughput dataset. It is microarray data on the transcriptional profile during sporulation in budding yeast. It contains more than 6,000 genes. Our evaluation includes clustering algorithm scalability pertaining to datasets with varying dimensions, the speedup factor, and the efficiency of the parallel model over the sequential implementation. Our experiments show that the gene expression data follow the pattern predicted in [Datta03] that is Diana appears to be solid performer also the group means for each cluster coincides with that in [Datta03]. We show that our parallel model is applicable to the clustering algorithms and more useful in applications that deal with high throughput data, such as gene expression data.

Genetic factors have been shown to be involved in etiology of a poor response to the antidepressant treatment with sufficient dosage and duration. Our goal was to identify the role of polymorphisms in the poor response to the treatment. To this end, 5 functional polymorphisms in 109 patients diagnosed with unipolar, major depressive disorder are analyzed. Due to the small sample size, exact conditional tests are utilized to analyze the contingency table. The data analysis involves: (1) Exact test for conditional independence in a high dimensional contingency table; (2) Marginal independence test; (3) Exact test for three-way interactions. The efficiency of program always limits the application of exact test. The appropriate methods for enumerating exact tables are the key to improve the efficiency of programs. The algorithm of enumerating the exact tables is also introduced.

Simultaneous measurement of the expression levels of thousands to ten thousand genes in multiple tissue types is a result of advancement in microarray technology. These expression levels provide clues about the gene functions and that have enabled better diagnosis and treatment of serious disease like cancer. To solve the mystery of unknown gene functions, biological to statistical mapping is needed in terms of classifying the genes. Here we introduce a novel approach of combining both statistical consistency and biological relevance of the clusters produced by a clustering method. Here we employ two performance measures in combination for measuring statistical stability and functional similarity of the cluster members using a set of gene expressions with known biological functions. Through this analysis we construct a platform to predict about unknown gene functions using the outperforming clustering algorithm.

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The minimum rank of a sign pattern matrix A is the minimum of the rank of the real matrices whose entries have signs equal to the corresponding entries of A. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly unrelated statements are established. For some special cases, such as when A is entrywise nonzero, or the minimum rank of A is at most 2, or the minimum rank of A is at least n - 1,(where A is mxn), the conjecture is shown to hold.Connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either -1 or 1 are explored. Sign patterns that almost require unique rank are also investigated.