Suppose two or more variables are jointly normally distributed. If there is a common relationship between these variables it would be very important to quantify this relationship by a parameter called the correlation coefficient which measures its strength, and the use of it can develop an equation for predicting, and ultimately draw testable conclusion about the parent population. This research focused on the correlation coefficient ρ for the bivariate and trivariate normal distribution when equal variances and equal covariances are considered. Particularly, we derived the maximum Likelihood Estimators (MLE) of the distribution parameters assuming all of them are unknown, and we studied the properties and asymptotic distribution of . Showing this asymptotic normality, we were able to construct confidence intervals of the correlation coefficient ρ and test hypothesis about ρ. With a series of simulations, the performance of our new estimators were studied and were compared with those estimators that already exist in the literature. The results indicated that the MLE has a better or similar performance than the others.

The arcsine distribution looks at the fraction of time one player is winning in a fair coin toss game and has been studied for over a hundred years. There has been little further work on how the distribution changes when the coin tosses are not fair or when a player has already won the initial coin tosses or, equivalently, starts with a lead. This thesis will first cover a proof of the arcsine distribution. Then, we explore how the distribution changes when the coin the is unfair. Finally, we will explore the distribution when one person has won the first few flips.

In recent years a new field of statistics has become of importance in many branches of experimental science. This is the Monte Carlo Method, so called because it is based on simulation of stochastic processes. By stochastic process, it is meant some possible physical process in the real world that has some random or stochastic element in its structure. This is the subject which may appropriately be called the dynamic part of statistics or the statistics of "change," in contrast with the static statistical problems which have so far been the more systematically studied. Many obvious examples of such processes are to be found in various branches of science and technology, for example, the phenomenon of Brownian Motion, the growth of a bacterial colony, the fluctuating numbers of electrons and protons in a cosmic ray shower or the random segregation and assortment of genes (chemical entities responsible for governing physical traits for the plant and animal systems) under linkage condition. Their occurrences are predominant in the fields of medicine, genetics, physics, oceanography, economics, engineering and industry, to name only a few scientific disciplines. The scientists making measurements in his laboratory, the meteriologist attempting to forecast weather, the control systems engineer designing a servomechanism (such as an aircraft or a thermostatic control), the electrical engineer designing a communication system (such as the radio link between entertainer and audience or the apparatus and cables that transmit messages from one point to another), economist studying price fluctuations in business cycles and the neurosurgion studying brain wave records, all are encountering problems to which the theory of stochastic processes may be relevant. Let us consider a few of these processes in a little more detail. In statistical physics many parts of the theory of stochastic processes were developed in correlation with the study of fluctuations and noise in physical systems (Einstein, 1905; Smoluchowski, 1906; and Schottky, 1918). Consequently, the theory of stochastic processes can be regarded as the mathematical foundation of statistical physics. The stochastic models for population growth consider the size and composition of a population which is constantly fluctuating. These are mostly considered by Bailey (1957), Bartlett (1960), and Bharucha-Reid (1960). In communication theory a wide variety of problems involving communication and/or control such as the problem of automatic tracking of moving objects, the reception of radio signals in the presence of natural and artificial disturbances, the reproduction of sound and images, the design of guidance systems, the design of control systems for industrial processes may be regarded as special cases of the following general problem; that is, let T denote a set of points in a time axis such that at each point t in T an observation has been made of a random variable X(t). Given the observations [x(t), t ϵT] and a quantity Z related to the observation, one desires to from in an optimum manner, estimates of, and tests of hypothesis about Z and various functions h(Z).

In evolutionary theory the existence of self-sacrificing cooperative traits poses a problem that has engendered decades of debate. The principal theories of the evolution of altruism are inclusive fitness, reciprocal altruism, and multilevel selection. To provide a framework for the unification o f these apparently disparate theories, this dissertation identifies two fundamental conditions required for the evolution of altruism: 1) non-zero-sum fitness benefits for cooperation and 2) positive assortment among altruistic behaviors. I demonstrate the underlying similarities in these three theories in the following two ways. First, I show that the game-theoretic model of the prisoner’s dilemm a (PD) is inherent to all three theories. While the PD has been used extensively to model reciprocal altruism, I demonstrate that the n-player PD captures fundamental aspects o f multilevel selection and inclusive fitness in that NPD model parameters relate simply to Simpson’s paradox, the Price covariance equation, and Hamilton’s rule. The tension between hierarchical levels that defines a PD reflects the tension between Abstract levels o f selection that is explicit in multilevel selection theory, and im plicit in the other two theories. Second, Ham ilton’s rule from inclusive fitness theory applies to the other theories. As mentioned, I demonstrate that this rule relates to multilevel selection via the NPD. I also show that Queller’s generalization of Hamilton’s rule applies to the conditional strategies of reciprocal altmism. This challenges the selfish-gene viewpoint by highlighting the fact that it is the phenotypes o f others, not their genotypes, that is critical to the evolution o f altruism. I integrate the PD and H am ilton’s rule as follows: the evolution o f altruism in general involves PD situations in which Hamilton’s rule specifies the necessary relationship between 1) the degree of non-zero-sumness within the PD and 2) the degree of positive assortment among altruistic behaviors. Additional contributions of this research include a demonstration that randomly formed associations can provide the necessary positive assortment for strong altruism to evolve, the development of a new selection decomposition that is symmetrical to the Price equation, and a game-theoretic analysis showing the essential similarity of weak and strong altruism under selection.

<p><br></p><p>How to set privacy parameters is a crucial problem for the consistent application of DP in practice. The current privacy parameters do not provide direct suggestions for this problem. On the other hand, different databases may have varying degrees of information leakage, allowing attackers to enhance their attacks with the available information. This dissertation provides an additional interpretation of the current DP notions by introducing a framework that directly considers the worst-case average failure probability of attackers under different levels of knowledge. </p><p><br></p><p>To achieve this, we introduce a novel measure of attacker knowledge and establish a dual relationship between (type I error, type II error) and (prior, average failure probability). By leveraging this framework, we propose an interpretable paradigm to consistently set privacy parameters on different databases with varying levels of leaked information. </p><p><br></p><p>Furthermore, we characterize the minimax limit of private parameter estimation, driven by $1/(n(1-2p))^2+1/n$, where $p$ represents the worst-case probability risk and $n$ is the number of data points. This characterization is more interpretable than the current lower bound $\min{1/(n\epsilon^2),1/(n\delta^2)}+1/n$ on $(\epsilon,\delta)$-DP. Additionally, we identify the phase transition of private parameter estimation based on this limit and provide suggestions for protocol designs to achieve optimal private estimations. </p><p><br></p><p>Last, we consider a federated learning setting where the data are stored in a distributed manner and privacy-preserving interactions are required. We extend the proposed interpretation to federated learning, considering two scenarios: protecting against privacy breaches against local nodes and protecting privacy breaches against the center. Specifically, we consider a non-convex sparse federated parameter estimation problem and apply it to the generalized linear models. We tackle two challenges in this setting. Firstly, we encounter the issue of initialization due to the privacy requirements that limit the number of queries to the database. Secondly, we overcome the heterogeneity in the distribution among local nodes to identify low-dimensional structures.</p>

<div>This work aims to provide solutions to two significant issues in the effective use and practical application of tensor completion as a machine learning method. The first solution addresses the challenge in designing fast and accurate recovery methods in tensor completion in the presence of highly sparse and highly missing data. The second takes on the need for robust uncertainty quantification methods for the recovered tensor.</div><div><br></div><div><b>Covariate-assisted Sparse Tensor Completion</b></div><div><b><br></b></div><div>In the first part of the dissertation, we aim to provably complete a sparse and highly missing tensor in the presence of covariate information along tensor modes. Our motivation originates from online advertising where users click-through-rates (CTR) on ads over various devices form a CTR tensor that can have up to 96% missing entries and has many zeros on non-missing entries. These features makes the standalone tensor completion method unsatisfactory. However, beside the CTR tensor, additional ad features or user characteristics are often available. We propose Covariate-assisted Sparse Tensor Completion (COSTCO) to incorporate covariate information in the recovery of the sparse tensor. The key idea is to jointly extract latent components from both the tensor and the covariate matrix to learn a synthetic representation. Theoretically, we derive the error bound for the recovered tensor components and explicitly quantify the improvements on both the reveal probability condition and the tensor recovery accuracy due to covariates. Finally, we apply COSTCO to an advertisement dataset from a major internet platform consisting of a CTR tensor and ad covariate matrix, leading to 23% accuracy improvement over the baseline methodology. An important by-product of our method is that clustering analysis on ad latent components from COSTCO reveal interesting and new ad clusters, that link different product industries which are not formed in existing clustering methods. Such findings could be directly useful for better ad planning procedures.</div><div><b><br></b></div><div><b>Uncertainty Quantification in Covariate-assisted Tensor Completion</b></div><div><br></div><div>In the second part of the dissertation, we propose a framework for uncertainty quantification for the imputed tensor factors obtained from completing a tensor with covariate information. We characterize the distribution of the non-convex estimator obtained from using the algorithm COSTCO down to fine scales. This distributional theory in turn allows us to construct proven valid and tight confidence intervals for the unseen tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, (2) it is data-driven and automatically adapts to unknown noise distributions and (3) in the high missing data regime, the inclusion of side information in the tensor completion model yields tighter confidence intervals compared to those obtained from standalone tensor completion methods.</div><div><br></div>

Percentiles are used everyday in descriptive statistics and data analysis. In real life, many quantities are normally distributed and normal percentiles are often used to describe those quantities. In life sciences, distributions like exponential, uniform, Weibull and many others are used to model rates, claims, pensions etc. The need to compare two or more independent populations can arise in data analysis. The ratio of percentiles is just one of the many ways of comparing populations. This thesis constructs a large sample confidence interval for the ratio of percentiles whose underlying distributions are known. A simulation study is conducted to evaluate the coverage probability of the proposed interval method. The distributions that are considered in this thesis are the normal, uniform and exponential distributions.

Surveys ordinarily ask questions in an ordinal scale and often result in missing data. We suggest a regression based technique for imputing missing ordinal data. Multilevel cumulative logit model was used with an assumption that observed responses of certain key variables can serve as covariate in predicting missing item responses of an ordinal variable. Individual predicted probabilities at each response level were obtained. Average individual predicted probabilities for each response level were used to randomly impute the missing responses using a uniform distribution. Finally, likelihood ratio chi square statistics was used to compare the imputed and observed distributions. Two other forms of multiple imputation algorithms were performed for comparison. Performance of our imputation technique was comparable to other 2 established algorithms. Our method being simpler does not involve any complex algorithms and with further research can potentially be used as an imputation technique for missing ordinal variables.

Testing the homogeneity of variances is an important problem in many applications since statistical methods of frequent use, such as ANOVA, assume equal variances for two or more groups of data. However, testing the equality of variances is a difficult problem due to the fact that many of the tests are not robust against non-normality. It is known that the kurtosis of the distribution of the source data can affect the performance of the tests for variance. We review the classical tests and their latest, more robust modifications, some other tests that have recently appeared in the literature, and use bootstrap and permutation techniques to test for equal variances. We compare the performance of these tests under different types of distributions, sample sizes and true ratios of variances of the populations. Monte-Carlo methods are used in this study to calculate empirical powers and type I errors under different settings.

For any company involved in sales, maximization of profit is the driving force that guides all decision-making. Many factors can influence how profitable a company can be, including external factors like changes in inflation or consumer demand or internal factors like pricing and product cost. Understanding specific trends in one's own internal data, a company can readily identify problem areas or potential growth opportunities to help increase profitability. In this discussion, we use an extensive data set to examine how a company might analyze their own data to identify potential changes the company might investigate to drive better performance. Based upon general trends in the data, we recommend potential actions the company could take. Additionally, we examine how a company can utilize predictive modeling to help them adapt their decision-making process as the trends identified from the initial analysis of the data evolve over time.