We consider the problem of maximum likelihood estimation of logistic sinusoidal regression models and develop some asymptotic theory including the consistency and joint rates of convergence for the maximum likelihood estimators. The key techniques build upon a synthesis of the results of Walker and Song and Li for the widely studied sinusoidal regression model and on making a connection to a result of Radchenko. Monte Carlo simulations are also presented to demonstrate the finite-sample performance of the estimators

Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}

In this thesis, we define differential operators for Hermitian Jacobi forms and Hermitian modular forms over the Gaussian number field Q(i). In particular, we construct Rankin-Cohen brackets for such spaces of Hermitian Jacobi forms and Hermitian modular forms. As an application, we extend Rankin's method to the case of Hermitian Jacobi forms. Finally we compute Fourier series coefficients of Hermitian modular forms, which allow us to give an example of the first Rankin-Cohen bracket of two Hermitian modular forms. In the appendix, we provide tables of Fourier series coefficients of Hermitian modular forms and also the computer source code that we used to compute such Fourier coefficients.

The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafé asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.

Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finally, we give upper and lower bounds for the expected rank in continuous time, and we show that the expected rank in the continuous time problem is at least as large as the normalized asymptotic expected rank in the full information discrete time version.

In Chapter 1 the classical secretary problem is introduced. Chapters 2 and 3 are variations of this problem. Chapter 2, discusses the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter p and finite time horizon n. The optimal strategy (continue or stop) depends on a sequence of threshold values (critical probabilities) which has an oscillating pattern. Several properties of this sequence have been proved by Dr. Allaart. Further properties have been recently proved. In Chapter 3, a gambler will observe a finite sequence of continuous random variables. After he observes a value he must decide to stop or continue taking observations. He can play two different games A) Win at the maximum or B) Win within a proportion of the maximum. In the first section the sequence to be observed is independent. It is shown that for each n>1, theoptimal win probability in game A is bounded below by (1-1/n)^{n-1}. It is accomplished by reducing the problem to that of choosing the maximum of a special sequence of two-valued random variables and applying the sum-the-odds theorem of Bruss (2000). Secondly, it is assumed the sequence is …

Receiver operating characteristic (ROC) analysis is one of the most widely used methods in evaluating the accuracy of a classification method. It is used in many areas of decision making such as radiology, cardiology, machine learning as well as many other areas of medical sciences. The dissertation proposes a novel nonparametric estimation method of the ROC surface for the three-class classification problem via Bernstein polynomials. The proposed ROC surface estimator is shown to be uniformly consistent for estimating the true ROC surface. In addition, it is shown that the map from which the proposed estimator is constructed is Hadamard differentiable. The proposed ROC surface estimator is also demonstrated to lead to the explicit expression for the estimated volume under the ROC surface . Moreover, the exact mean squared error of the volume estimator is derived and some related results for the mean integrated squared error are also obtained. To assess the performance and accuracy of the proposed ROC and volume estimators, Monte-Carlo simulations are conducted. Finally, the method is applied to the analysis of two real data sets.

Semi-supervised learning (SSL) is the most practical approach for classification among machine learning algorithms. It is similar to the humans way of learning and thus has great applications in text/image classification, bioinformatics, artificial intelligence, robotics etc. Labeled data is hard to obtain in real life experiments and may need human experts with experimental equipments to mark the labels, which can be slow and expensive. But unlabeled data is easily available in terms of web pages, data logs, images, audio, video les and DNA/RNA sequences. SSL uses large unlabeled and few labeled data to build better classifying functions which acquires higher accuracy and needs lesser human efforts. Thus it is of great empirical and theoretical interest. We contribute two SSL algorithms (i) adaptive anomaly detection (AAD) (ii) hybrid anomaly detection (HAD), which are self evolving and very efficient to detect anomalies in a large scale and complex data distributions. Our algorithms are capable of modifying an existing classier by both retiring old data and adding new data. This characteristic enables the proposed algorithms to handle massive and streaming datasets where other existing algorithms fail and run out of memory. As an application to semi-supervised anomaly detection and for experimental illustration, we …