This is an advanced introduction to regression modeling and prediction, including traditional and modern computationally-intensive methods. The following topics will be covered: (1) Theory and practice of linear models, including the relevant distribution theory, estimation, confidence and prediction intervals, testing, model and variable selection, generalized least squares, robust fitting, and diagnostics; (2) Generalized linear models, including likelihood formulation, estimation and inference, diagnostics, and analysis of deviance; and (3) Large and small-sample inference as well as inference via the bootstrap, cross-validation, and permutation tests. (4 Credits)
Prerequisites: Knowledge of linear algebra; Knowledge of regression and analysis of variance at the level of STATS 500; Knowledge of probability and statistical theory at the level of BIOSTAT 601/602.
Statistics 607 (I/II): Programming and Numerical Methods in Statistics (Offered prior to Fall 2019)
This course is an advanced introduction to modern programming (Part I) and numerical analysis (Part II) techniques used in statistics, modeling and data analysis. Part I course topics include: Basic data structures, structured data formats, iteration and recursion, functional programming, classes and object-oriented programming, memory management, strategies for documenting, and debugging code. This part of the course will cover programming fundamentals relevant for research on statistical methodology, and for working with large and complex data sets. (1.5 Credits)
Part II course topics include: sorting and binary searches, root finding in one dimension, interpolation techniques, low-dimensional numerical integration, solving triangular systems, basic matrix factorizations (LU, Cholesky, QR), Schur and singular value decompositions, sparse matrices. This part of the course will cover elementary algorithms that are useful for numerical analysis and programming with data.
Statistics 608 (I/II): Methods in Optimization Statistics (Offered prior to Fall 2019)
This course is an advanced introduction to deterministic (Part I) and stochastic (Part II) optimization techniques. Part I course topics include: basic result's from mathematical analysis, role of convexity in optimization, Karush-Kuhn-Tucker conditions in constrained optimization, majoration algorithms and their applications (EM algorithm), Newton's method and extensions, convergence results, convex programming and duality. The material covers both theoretical and implementation issues, as well as application to statistical models. (1.5 Credits)
Part II course topics include: basic Monte Carlo methods (random number generators, variance reductions techniques), an introduction to Markov chains (irreducibility, recurrence, ergodicity), Markov Chain Monte Carlo methods (Metropolis-Hastings and Gibbs sampling algorithms, data-augmentation techniques, convergence diagnostics), and stochastic optimization (simulated annealing and stochastic approximation). This part of the course covers both theory and applications to complex statistical models.
Advisory Pre-requisites: MATH 451, STATS 425, STATS 426. Computer programming experience recommended. Department Consent Required.
Stats 610: Statistical Inference (Last offered Fall 2019)
This course introduces students to the theory of statistical inference. It starts with a review of topics in probability theory including densities, expectation, random vectors and covariance matrices, independence, and conditioning. It then introduces exponential families and sufficiency and develops the theory of point estimation including unbiased and Bayesian estimation, conditional distributions, variance bounds and information. The theory of hypothesis testing is also covered, including uniformly most powerful tests and the duality between testing and interval estimation. Additional topics that may be covered include curved exponential families, equivariant estimation, and empirical Bayes and shrinkage estimators. (3 Credits)
Prerequisites: MATH 451, STATS 425, and STATS 426 or equivalent courses in probability, statistics and real analysis.
Stats 611: Large Sample Theory (Last offered Winter 2020)
This course covers topics in large sample theory that are central for statistical inference, including: (1) modes of convergence, central limit theorems for averages and medians, and asymptotic relative efficiency; (2) estimating equations including the law of large numbers for random functions, consistency and asymptotic normality for maximum likelihood and M-estimators, the E-M algorithm, and asymptotic confidence intervals; (3) large sample theory for likelihood ratio tests. In addition, simultaneous inference and nonparametric regression are also covered. Other possible topics include theory for two-sided tests and tests in higher dimensions. (3 Credits)
Prerequisites: STATS 610
Stats 621: Theory of Probability II (Last offered Fall 2019)
This course is an introduction to measure-theoretic probability theory, with emphasis on rigorous treatment of the various topics discussed in the course. Topics to be covered include: (i) constructions of probability spaces, Kolmogorov's consistency theorem; independence of families of random variables, Borel-Cantelli lemmas and 0-1 laws; (ii) various modes of convergence (in probability, almost surely, in Lp, in distribution) and properties of weak convergence, (iii) laws of large numbers, (iv) central limit theorems for sequences and triangular arrays, (v) conditional expectations and distributions and (vi) discrete time martingale theory. In addition, Brownian motion, continuous time martingales and elements of ergodic theory may be covered. (3 Credits)
Prerequisites: STATS 520 or equivalent course in measure theory, STATS 620.