Statistics

Design and analysis of sample surveys. Sources of error; questionaire design; simple random, stratified, systematic, and cluster sampling; plus practical application of these concepts.

Probability, counting rules, conditional probablity, independence, Bayes’ Rule. Discrete and continuous distributions, means, variances, moment generating functions. Multivariate probability distributions, marginal and conditional distributions, covariance. Distributions of functions of random variables.

Mathematical foundations of point estimation, confidence intervals, tests of hypotheses, linear models and analysis of variance.

Intended for graduate students not majoring in statistics. Description and inference using proportions and odds ratios, multi-way contingency tables, logistic regression and other generalized linear models, and loglinear models applications.

Introduction to nonparametric statistics, including one and two sample testing and estimation methods, one- and two-way layout models, and correlation and regression models.

Review of matrix theory, univariate normal, t, chi-squared, F and multivariate normal distributions. Inference about multivariate means including Hotelling’s T squared, multivariate analysis of variance, multivariate regression and multivariate repeated measures. Inference about covariance structure including principal components, factor analysis and cannonical correlation. Multivariate classification techniques including discriminant and cluster analysis. Additional topics at the discretion of the instructor, time permitting.

Introduction to communication, management, organizational, computational, and statistical thinking skills necessary to consulting in statistics. Integration of graphical and numerical computing tools, research design concepts, data summary and statistical inference methods.

Descriptive statistics, estimation, significance tests, two-sample comparisons, methods for nominal and ordinal data, regression and correlation, introduction to multiple regression.

Further topics in multiple regression, model building, analysis of variance, analysis of covariance, multivariate analysis of categorical data.

Statistical inference based on t, F, and X2 tests. Analysis of variance for basic experimental designs. Factorial experiments. Regression analysis and analysis of covariance.

Analysis of split-plot and nested designs with incomplete blocks, confounding and fractional replications. Analysis of count data. Nonparametric methods.

Analysis of epidemiological studies, measures of morbidity and mortality, methods for rates and proportions, bioassay, longitudinal data analysis.

Survival analysis, Kaplan-Meier estimates, proportional hazards model.

Simple linear regression; multiple regression; model selection residual analysis; influence diagnostics; multicollinearity; ANOVA and regression; generalized linear models; nonlinear regression.

Overview of normal theory inference, nonparametric, and categorical data methods; basic concepts of experimental design; analysis of variance; introduction to factorial and nested experiments.

Theory for analysis of linear models in univariate data; distributions of quadratic forms; full rank linear models; fixed effect models of less than full rank; balanced random and mixed models; unbalanced random and mixed models.

Theory of probability. Probability spaces, continuous and discrete distributions, functions of random variables, multivariate distributions, expectation, conditional expectation, central limit theorem, useful convergence results, sampling distributions, distributions of order statistics, empirical distribution function.

Estimation and hypothesis testing. Sufficiency, information, estimation, maximum likelihood, confidence intervals, uniformly most powerful tests, likelihood ratio tests, sequential testing, univariate normal inference, decision theory, analysis of categorical data.

Varieties of categorical data, cross-classification tables, tests for independence. Measures of association. Loglinear models for multi-dimensional tables. Logit models and analogies with regression. Specialized methods for ordinal data.

Inference based on rank statistics– one, two and k-sample problems, correlation and regression problems and analysis of contingency tables. Conditionally distribution-free rank tests. Pitman asymptotic relative efficiency.

Techniques for analyzing multivariate data. Emphasis on MANOVA and tests on the structure of the dispersion matrix. Topics will include discriminant, factor, profile, and cluster analyses.

Random-variable generation, accept-reject methods. Monte Carlo Optimization, Markov Chain Methods, diagnostic tools, programming in R and WinBugs. Provide an introduction to statistical methods based on Monte Carlo Methods which have become a standard for statisticians today.

Special topics designed to meet the needs and interests of individual students.

Special topics courses in statistics.

Special topics of an advanced nature suitable for seminar treatment but not given in regular courses.

Supervised statistical consulting involving the planning and/or analysis of research data. Whenever possible, the student will meet with the researcher. Supervision by a faculty member or delegated authority. Post internship report is required.

Theoretical development of statistical methods for analyzing life history data, including censored data and truncated data. Topics covered include: Kaplan-Meier estimator, k-sample tests, proportional hazards regression, and the asymptotic theory associated with all these. Throughout, the counting process approach to survival analysis will be used. A student who takes this course should have had a year-long sequence in probability that covers martingales and the Lindeberg-Feller theorem.

STA 7233 Advanced Regression

Fitting of generalized linear models, diagnostics, asymptotic theory, overdispersion, estimating equations, mixed models, generalized additive models, smoothing.

Review of different models of convergence. Cramer-Wold device. Multivariate CLT. Asymptotic theory of empirical distribution and sample quantiles. Bahadur’s representation. Asymptotic theory of sample moments. Delta method and its multiparameter generalization. Variance stabilizing transformation. U-statistics: asymptotic theory and its statistical applications. Hoeffding’s decomposition. Asymptotic theory of maximum likelihood estimation. Wald’s consistency theorem for MLE. Asymptotic normality and efficiency. Asymptotic theory of GLRTs. Statistical applications: asymptotic theory or categorical data, linear models, and generalized linear models.

Decision rules and risk functions. Sufficiency, Minimax, and Bayes rules for estimation of location and scale parameters.

Bayesian statistical inference. Inference using large samples. Relative efficiencies of tests and estimates with special reference to Pitman and Bahadur efficiencies.

The objective of the Bayesian Theory course is to provide students with a solid foundation of the theory underlying the Bayesian paradigm. In particular, we will discuss issues related to selection of priors, Bayesian inference both exact and asymptotic, Bayesian model selection, high dimensional problems, and if time permits, some issues related to Bayesian robustness.

Measure and probability spaces; random variables; distribution functions; abstract Lebesgue and Stieltjes integration; monotone; dominated, Cauchy, and mean convergence; Fubini and Radon-Nikodym theorems; zero-one laws.

Summability of independent random variables, laws of large numbers, convergence in distribution, characteristic functions, uniqueness and continuity theorems, the Lindeberg-Feller central limit theorem, degenerate convergence criterion.

Theoretical foundations of nonparametric statistics: theory of U-statistics, Noether’s theorem and Pitman asymptotic relative efficiency, estimation and hypothesis testing with one and two sample models, theory of linear rank statistics, applications to general linear models analyses.⁸

Topics to be covered may include: branching processes, Brownian motion, continuous state space Markov chains, diffusion processes, Markov chain Monte Carlo, martingales,point processes, renewal processes, stationary processes, stochastic calculus, stochastic dfferential equations.

Possible Topics: Smoothing Methods, Analysis of Longitudinal Data, Data Mining and Statistical Learning, Mixed Models, Theory and Methods, Resampling Methods, Functional Data Analysis.⁹

Research for doctoral students before admission to candidacy. Designed for students with a master’s degree in the field of study or for student who have been accepted for a doctoral program. Not open to students who have been admitted to candidacy.