205 Probability and Measure

Probability and Measure. Introduction to probability spaces, the theory of measure and integration, random variables, and limit theorems. Distribution functions, densities, and characteristic functions; convergence of random variables (a.s., L^p, in probability) and their distributions; uniform integrability and the Lebesgue convergence theorems. Weak and strong laws of large numbers, central limit theorem, conditional probability and expectation and their statistical relevance. Martingales, Sequential statistical tests, introduction to Brownian motion.

3 units. Prerequisite: MTH 203 or equivalent.

Typical Text: Billingsley, Probability and Measure (parts of chap. 1-6).

213 Introduction to Statistical Methods

Formal introduction to basic theory and methods of probability and statistics: probability and sample spaces, independence, conditional probability and Bayes' theorem; random variables, distributions, moments and transformations. Parametric families of distributions and central limit theorem. Sampling distributions, traditional methods of estimation and hypothesis testing. Elements of likelihood and Bayesian inference. Basic discrete and continuous statistical models.

Provides an introduction to the basic ideas of probability and statistics, building a mathematical foundation sufficient for virtually all courses at the graduate level. Students must have acquired facility with multivariate calculus prior to taking the course. The major elements of probability and mathematical statistics are presented, with an emphasis on a thorough understanding of the ideas. The first part of the course covers topics in probability with an emphasis on those needed in statistics. The second part of the course provides a foundation in both Bayesian and classical inference.

3 units. Prerequisite: MTH 103 or equivalent (may be taken concurrently).

Typical Text: Casella and Berger, Statistical Inference (chap. 2-6)

214 Probability and Statistical Models

This course covers a range of theory, modelling methodology and computational topics arising in probability and statistical analysis: Theoretical and simulation aspects of multivariate distribution theory including multivariate normal modelling and other parametric families. Development and use of simulation methodology and applied probability models in statistical analysis, including the generation of random variables with specified distributions, the Monte Carlo method and Monte Carlo integration. Extensive development of Markov Chain Monte Carlo methods. Elements of applied stochastic processes including Markov process theory, linear systems theory, and ARMA models. Latent variable probability models including mixture models, hidden Markov models and missing data problems. Aspects of multinormal/linear statistics, including normal/Wishart distribution theory, regression and graphical/network models. Associated methods of linear algebra and probability calculus. Statistical computing using Matlab or R.

3 units. Prerequisites: STA 215, 244 and STA 290, or close equivalents. This is a second level statistics graduate course that relies in the background in core mathematical statistics, Bayesian and non-Bayesian inference, applied modelling in statistics and computation in these first level graduate courses.

Typical text: Course notes and computational resources of the instructor(s). Useful support texts include: (a) Gamerman & Lopes, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, (b) Robert & Casella, Monte Carlo Statistical Methods. Other texts and support materials will be introduced as needed.

215 Statistical Inference

Comparison of classical, likelihood and Bayesian approaches to statistical inference. Foundations of point and interval estimation, and properties of estimators (bias, consistency, efficiency, sufficiency, robustness). Testing: Type I and II errors, power; likelihood ratios; Bayes factors, posterior probabilities of hypotheses, the predictivist perspective, the likelihood principle. Estimation and testing in exponential families. Principles underlying model selection and comparison.

This course takes an advanced and rigorous look at mathematical statistics and approaches to inference. In addition to covering central concepts and models of statistics, differing philosophical perspectives on scientific inference are discussed and compared.

3 units. Prerequisite: STA 213 or equivalent.

Typical text: Casella and Bergen, Statistical Inference. Other texts and support material introduced as needed.

216 Generalized Linear Models

Generalized Linear Models.  Models for discrete data, nonlinear hierarchical models.

Likelihood-based inference in generalized linear models (GLMs). Brief review of multiple linear regression, theory and practice. Theory of GLMs. Discrete models: binary regression and contingency tables. Introduction to log-linear models. Data analysis: model fitting, model choice, and residuals-based diagnostics. Elements of sampling theories and Bayesian theories of inference in GLMs. Routine use of statistical software, for model exploration and fitting and for data analysis. MCMC methods for Bayesian analysis in GLMs. Failure-time data (survival analysis); extensions (as time permits) to include growth curves and dynamic linear models, repeated categorical data, frailty models, quasilikelihood and generalized estimating equations (GEEs)

This course covers the theory and practice of modern regression modeling within the unifying framework of the GLM. Students will use appropriate computer software for data analysis and manipulation, model construction, assessment and model-based inference.

3 units. Prerequisite: STA 215 and STA 244 or equivalent.

Typical texts: McCullagh & Nelder, Generalized Linear Models; Chambers & Hastie, Statistical Models in S; Lindsey, Applying Generalized Linear Models. Software may include R, Matlab, BUGS, or SAS.

244 Linear Models

Linear regression. Normal linear models, ANOVA, diagnostics, hierarchical linear models, Bayesian analysis. Multiple linear regression and model building. Exploratory data analysis techniques, transformations of dependent and independent variables, variable selection, parameter estimation and interpretation, prediction, Bayesian hierarchical linear models, Bayes factors and intrinsic Bayes factors for linear models, and Bayesian model averaging.

This course investigates the essential concepts of linear models from both Bayesian and classical viewpoints. Selected topics in Markov chain Monte Carlo simulation will be introduced as required.

3 units. Prerequisite: STA 213 or equivalent.

Typical texts: Weisberg, Applied Linear Regression; Gelman et al., Statistical Models in S.

253 (Mth 216) Applied Stochastic Processes

Finite Markov chains - long-range behavior and invariant probability, classification of states, return times, transient states. Countable Markov chains -- recurrence and transience, branching process, continuous-time Markov chains. Poisson process, finite state space, birth-and-death processes, Optimal and optional stopping. Martingales and martingale convergence. Renewal processes, M/G/1 and G/M/1 queues. Reversible Markov chains and processes. Brownian motion -- recurrence and transience, fractal nature of Brownian motion, Brownian motion with drift. Introduction to stochastic integration, Ito's formula. Simulation.

3 units. Prerequisites: STA 213 or equivalent.

Typical text: Lawler, Introduction to Stochastic Processes.

290 Modern Statistical Data Analysis

This course provides a broad introduction to a range of topics in statistical science, data analysis, modelling and statistical practice and reporting. Topics will be selected from:

• data types, data manipulation and analysis, including ranges of data sets from a variety of application fields
• statistical computing, data analysis and graphics using elements of the R programming environment and a brief introduction to C programming
• methods of exploratory data analysis and statistical graphics
• examples of problems of binary and count/categorical data, regression problems and data, and others
• elements of statistical inference using probability models, including basic issues of sampling-theory and Bayesian inference, especially in the context of inference in problems of proportions
• modeling "related" phenomena through the notion of hierarchical models
• statistical and mathematical techniques including elements of simulation and linear/matrix algebra useful in statistics
• use of unix tools, including emacs editors, TeX and LaTeX, the R library.
• readings from various texts, research and teaching literature

Though the course does not include rigourous development of statistical theory and methods, we will use and review various concepts and methods of inference, so that some familiarity with basic statistics desirable. Co-registration in STA 213 or recent experience with similar courses will be most useful.

3 units. No prerequisites, though some familiarity with basic statistics desirable. Co-registration in STA 213 useful.

Typical Text:  No fixed text, but sources such as Venables & Ripley, Modern Applied Statistics with S-Plus will be useful. Extensive course notes are available (see current course web site).

376 Advanced Modelling and Scientific Computation

Computing. numerical integration, advanced random number generation and Monte Carlo methods, including convergence and limit results for MCMC.

Techniques for maximization and integration suitable for computations in statistical analyses and particularly Bayesian inference. Quadrature methods; Simulated annealing; Metropolis techniques; methods of Monte-Carlo and Gibbs sampling; error analysis, variance reduction, and comparisons with standard derivative-based optimization methods and quadrature-based integration methods.

Advanced statistical modeling and modern numerical methods useful in implementing statistical procedures for data analysis, model exploration, inference, and prediction. Methods are applied to substantial problems in discrete multivariate analysis, time series, econometrics, non-linear regression models, density estimation, applications with censored and missing data, hierarchical models, mixture modeling, and non-linear regressions.

3 units. Prerequisite: STA 215 and STA 214 or equivalent.

Material will be drawn from many sources, such as B. Ripley, Stochastic Simulation, M. Tanner, Tools for Statistical Inference, Gilks, Richardson and Spiegelhalter, Markov Chain Monte Carlo in Practice, and the current research literature.

390 Statistical Consulting Workshop

Tutorials and workshops on statistical consulting. Students address and solve consulting problems submitted to Duke Statistics's campus-wide consulting program. May be taken more than once.

1 unit. Consent of instructor required.

293, 294 Special Topics in Statistics

Selected topics not covered in core courses, more advanced topics, and topics involving current research directions in statistical science and related areas.

3 units. Prerequisite: STA 213 or consent of instructor. More advanced statistics courses, including STA 214, 215 and 244, preferred.

356 Time Series and Forecasting

Time series data and models: trend, seasonality, and regressions. Traditional models: EWMA, EWR, ARMA. Dynamic linear models (DLMs). Bayesian learning, forecasting, and smoothing. Mathematical structure of DLMs and related models. Intervention, forecast monitoring, and control. Structural change in time series. Multiprocess models and mixture analysis. Multivariate models, constrained and aggregate forecasting, and forecast combination. Applications using computer software. Other topics, including spectral analysis, as time permits.

3 units. Prerequisite: STA 215 or equivalent.

Typical Text: M. West & P.J. Harrison, Bayesian Forecasting & Dynamic Models, and supplementary class notes and readings.

357 Stochastic Processes

Conditional probabilities and Radon-Nikodym derivatives of measures, tightness and weak convergence of probability measures, measurability and observability. Markov chains, Brownian motion, Poisson processes. Gaussian processes, birth-and-death processes, and an introduction to continuous-time martingales.

3 units. Prerequisites: STA 205 (or MTH 290) and STA 215.

395 Readings in Statistical Science

Informal advanced seminar course on topics related to current research in Duke Statistics and research frontiers in statistical science and interdisciplinary applications.

Variable credit. Consent of instructor required.

Additional Courses:

217 Ordinal Data Modeling
218 Statistical Data Mining
226 Statistical Decision Theory
270 Statistical Methods for Computational Biology
280 Spatial Statistics
281 Modern Nonparametric Theory and Methods