Statistics and Probability Commons^™

Nonparametric Derivative Estimation Using Penalized Splines: Theory And Application, Bright Antwi Boasiako

Doctoral Dissertations

This dissertation is in the field of Nonparametric Derivative Estimation using
Penalized Splines. It is conducted in two parts. In the first part, we study the L2
convergence rates of estimating derivatives of mean regression functions using penalized splines. In 1982, Stone provided the optimal rates of convergence for estimating derivatives of mean regression functions using nonparametric methods. Using these rates, Zhou et. al. in their 2000 paper showed that the MSE of derivative estimators based on regression splines approach zero at the optimal rate of convergence. Also, in 2019, Xiao showed that, under some general conditions, penalized spline estimators …

Model-Free Descriptive Modeling For Multivariate Categorical Data With An Ordinal Dependent Variable, Li Wang

Doctoral Dissertations

In the process of statistical modeling, the descriptive modeling plays an essential role in accelerating the formulation of plausible hypotheses in the subsequent explanatory modeling and facilitating the selection of potential variables in the subsequent predictive modeling. Especially, for multivariate categorical data analysis, it is desirable to use the descriptive modeling methods for uncovering and summarizing the potential association structure among multiple categorical variables in a compact manner. However, many classical methods in this case either rely on strong assumptions for parametric models or become infeasible when the data dimension is higher. To this end, we propose a model-free method …

Joint Asymptotics For Smoothing Spline Semiparametric Nonlinear Models, Jiahui Yu

Doctoral Dissertations

We study the joint asymptotics of general smoothing spline semiparametric models in the settings of density estimation and regression. We provide a systematic framework which incorporates many existing models as special cases, and further allows for nonlinear relationships between the finite-dimensional Euclidean parameter and the infinite-dimensional functional parameter. For both density estimation and regression, we establish the local existence and uniqueness of the penalized likelihood estimators for our proposed models. In the density estimation setting, we prove joint consistency and obtain the rates of convergence of the joint estimator in an appropriate norm. The convergence rate of the parametric component …

Asymptotic Behavior Of The Random Logistic Model And Of Parallel Bayesian Logspline Density Estimators, Konstandinos Kotsiopoulos

Doctoral Dissertations

This dissertation is comprised of two separate projects. The first concerns a Markov chain called the Random Logistic Model. For r in (0,4] and x in [0,1] the logistic map f_r(x) = rx(1 - x) defines, for positive integer t, the dynamical system x_r(t + 1) = f(x_r(t)) on [0,1], where x_r(1) = x. The interplay between this dynamical system and the Markov chain x_r,N(t) defined by perturbing the logistic map by truncated Gaussian noise scaled by N^-1/2, where N -> infinity, is studied. A natural question is …

Inference From Network Data In Hard-To-Reach Populations, Isabelle Beaudry

Doctoral Dissertations

The objective of this thesis is to develop methods to make inference about the prevalence of an outcome of interest in hard-to-reach populations. The proposed methods address issues specific to the survey strategies employed to access those populations. One of the common sampling methodology used in this context is respondent-driven sampling (RDS). Under RDS, the network connecting members of the target population is used to uncover the hidden members. Specialized techniques are then used to make inference from the data collected in this fashion. Our first objective is to correct traditional RDS prevalence estimators and their associated uncertainty estimators for …

Advanced Sequential Monte Carlo Methods And Their Applications To Sparse Sensor Network For Detection And Estimation, Kai Kang

Doctoral Dissertations

The general state space models present a flexible framework for modeling dynamic systems and therefore have vast applications in many disciplines such as engineering, economics, biology, etc. However, optimal estimation problems of non-linear non-Gaussian state space models are analytically intractable in general. Sequential Monte Carlo (SMC) methods become a very popular class of simulation-based methods for the solution of optimal estimation problems. The advantages of SMC methods in comparison with classical filtering methods such as Kalman Filter and Extended Kalman Filter are that they are able to handle non-linear non-Gaussian scenarios without relying on any local linearization techniques. In this …

Mixture Of Factor Analyzers With Information Criteria And The Genetic Algorithm, Esra Turan

Doctoral Dissertations

In this dissertation, we have developed and combined several statistical techniques in Bayesian factor analysis (BAYFA) and mixture of factor analyzers (MFA) to overcome the shortcoming of these existing methods. Information Criteria are brought into the context of the BAYFA model as a decision rule for choosing the number of factors m along with the Press and Shigemasu method, Gibbs Sampling and Iterated Conditional Modes deterministic optimization. Because of sensitivity of BAYFA on the prior information of the factor pattern structure, the prior factor pattern structure is learned directly from the given sample observations data adaptively using Sparse Root algorithm. …

Measuring Inequality: Statistical Inference Theory With Applications, Mihaela Paun

Doctoral Dissertations

In this dissertation we develop statistical inference for the Atkinson index, one of the measures of inequality used in studying economic inequality.

Specifically, we construct empirical estimators for the Atkinson index, both in the parametric and nonparametric case, and derive formulas for the asymptotic variances for the estimators. These statistics are used for testing hypothesis and constructing confidence intervals for the Atkinson index. We test the validity and the robustness of the asymptotic theory, by simulations (using R, a language and environment for statistical computing and graphics), in the case of one and two populations. In addition to proving asymptotic …