Core Ph.D Courses
STAT GR6101x. Statistical Modeling and Data Analysis I. 4 pts.
The first semester of a 2 semesters sequence in applied statistics for first year doctoral students in Statistics. Statistical methods including linear and generalized regression; random-effects models; methods for categorical data; survival analysis; and nonparametric methods. The use of statistical packages. Modeling. Exploratory data analysis; modern nonparametric regression. Restricted to students in the doctoral program in Statistics.
STAT GR6102y. Statistical Modelling and Data Analysis II. 4 pts. Prerequisites:STAT GR6101
Continuation of GR6101.
STAT GR6103y. Statistical Modeling and Data Analysis III. 3 pts. Prerequisites: Intended primarily for doctoral students in Statistics.
Spatial statistical, HMMs, dynamic programming convex optimaztion, numerical linear algebra for statistical applications, grid computing, SDEs.
STAT GR6105x. Probability Theory I. 4 pts. Prerequisites: A thorough knowledge of elementary real analysis and some previous knowledge of probability.
Overview of measure and integration theory. Probability spaces and measures, random variables and distribution functions. Independence, Borel-Cantelli lemma, zero-one laws. Expectation, uniform integrability, sums of independent random variables, stopping times, Wald’s equations, elementary renewal theorems. Laws of large numbers. Characteristic functions. Central limit problem; Lindeberg-Feller theorem, infinitely divisible and stable distributions. Cramer’s theorem, introduction to large deviations. Law of the iterated logarithm, Brownian motion, heat equation. For students in the doctoral program in Statistics only. Students in a masters program must seek permission from the Director of the MA program in Statistics; students in an undergraduate program must seek permission from the Director of Undergraduate Studies in Statistics.
STAT GR6106y. Probability Theory II. 4 pts. Prerequisites: Statistics GR6105.
Conditional distributions and expectations. Martingales; inequalities, convergence and closure properties, optimal stopping theorems, Burkholder-Gundy inequalities, Doob-Meyer decomposition, stochastic integration, Ito’s rule. Brownian motion: construction, invariance principles and random walks, study of sample paths, martingale representation results Girsanov Theorem. The heat equation, Feynman-Kac formula. Dirichlet problem, connections with potential theory. Introduction to Markov processes: semigroups and infinitesimal generators, diffusions, stochastic differential equations. For students in the doctoral program in Statistics only. Students in a masters program must seek permission from the Director of the MA program in Statistics; students in an undergraduate program must seek permission from the Director of Undergraduate Studies in Statistics.
STAT GR6107x. Theory of Statistical Inference I. 4 pts.
A general introduction to mathematical statistics and statistical decision theory. Elementary decision theory, Bayes inference, Neyman-Pearson theory, hypothesis testing, most powerful unbiased tests, confidence sets. Estimation: methods, theory, and asymptotic properties. Likelihood ratio tests, multivariate distribution. Elements of general linear hypothesis, invariance, nonparametric methods, sequential analysis. For students in the doctoral program in Statistics only. Students in a masters program must seek permission from the Director of the MA program in Statistics; students in an undergraduate program must seek permission from the Director of Undergraduate Studies in Statistics.
STAT GR6108y. Theory of Statistical Inference II. 4 pts. Prerequisites:STAT GR6107
Continuation of STAT GR6107. For students in the doctoral program in Statistics only. Students in a masters program must seek permission from the Director of the MA program in Statistics; students in an undergraduate program must seek permission from the Director of Undergraduate Studies in Statistics.
STAT GR6110x. Probability Theory III. 3 pts. Prerequisites: Stat GR6105 and GR6106
This course is intended to follow two semesters of graduate measure theoretic probability theory. The topics to be covered will include the following, but may be slightly changed from year to year depending on the preferences of the professor and also of the students: Advanced stochastic Integration for Semimartingales; Stochastic Differential Equations and Markov Processes; Connections to Partial Differential Equations. An introduction to the General Theory of Mathematical Finance. The Modern Theory of the Discretization of Processes and Statistical Methods for Volatility Estimation within an Ito Process Context.
STAT GR6105x and y. Statistical Consulting. 2 pts. Prerequisites: instructor’s permission. Repeated enrollment for credit is permitted.
Available to SSP, SMP
STAT G9001x-G9002y. Seminar In Advanced Mathematical Statistics. 3 pts.
Departmental seminar in Statistics.
STAT G9003x-G9004y. Seminar In Advanced Probability.. 3 pts.
Departmental seminar in probability theory.
Ph.D Seminar Courses
STAT GR8220y. Topics in Spatial Statistics. 3 pts.
This course will provide an overview of some statistical methods for spatial data. The focus will be mainly on spatial point processes and Gaussian random fields. We will study descriptive methods for spatial point data, models for spatial point processes and methods for fitting them to data. We will also look at inhomogeneous point processes, and bootstrap methods. For Gaussian random fields, we will discuss properties such as stationarity and isotropy. We will cover variogram estimation and fitting, kriging, fixed-and increasing-domain asymptotics, as well as methods for modeling non-stationary data. Examples will be taken from journal articles and various research projects covering a variety of fileds, such as astronomy, syndromic surveillance, epidemiology and fMRI.
STAT GR8243x and y. Topics In Advanced Probability. 3 pts. Prerequisites: Statistics GR6106 and Department’s permission
Available to SSP, SMP Topic to vary each time course is given. May be repeated for credit.
STAT GR8263y. Stochastic Differential Equations and Applications. 3 pts.
Prerequisite: Statistics GR6105-GR6106 or member of the Department?s permission. Elements of the general theory of stochastic processes; Martingales; Doob-Meyer decomposition; Strochastic integrals; differential equations; existence and uniqueness of strong and weak partial differential equations; comparison and approximation theorems; Stroock-Varadhan theory; applications to stochastic optimization.
STAT GR8273x. Topics in the Mathematics of Finance. 3 pts. Prerequisites: Math G4161/Stat GR6105 and Stat GR6106 or instructor’s permission.
Notions of arbitrage, completeness, hedging, and equivalent martingale measures for financial markets. Pricing of contingent claims and optimal portfolios, for both complete and incomplete markets. Equilibrium; the Capital Asset Pricing model. Market frictions: portfolio constraints, transaction costs.
STAT GR8285y. Statistical Analysis and Modeling of Neural Spike Train Data. 3 pts.
We will discuss a variety of statistical methods for analyzing and modeling spike train data from single and multiple neurons. Statistical topics to be covered include: point proceses; generalized linear models; regularization; state space models; Kalman and particle filtering; the Expectation-Maximization algorithm; and optimiztion/convexity techniques. The course will be structured around applications to neural data, including: spike-triggered averaging; estimation of neural encoding models that inlcude spike-history and multineuronal interaction effects; estimating integrate-and-fire models from extracellular and intracellular data; performing optimal smoothing of noisy voltage- and calcium-imaging data; decoding spike trains; and estimating mutual information between stimuli and neutral responses.
STAT GR8305x or y. Advanced Topics in Survival Analysis. 3 pts. Prerequisites:STAT GR6107-GR6108 or instructor’s permission
Counting process representations for the Kaplan-Meier estimator and the maximum Cox partial likelihood estimator; semi-parametric efficiency bounds; censored multivariate outcomes; truncation; stratified analysis; survival analysis in epidemiology.
STAT GR8315y. Analyzing Large Correlated Data with Applications in Genetics and Classifications. 3 pts. Prerequisites: None.
This topics course presents recently developed tools for handling large correlated data sets, focusing on applications in human genetics, microarray analysis, and machine learning.
STAT GR8318x or y. Semiparametric Inference. 3 pts. Prerequisites: Instructor’s permission.
Inference for Euclidean parameters in the presence of infinite dimensional nuisance parameters. Tangent space and asymptotic information bounds, construction of estimates, examples.
STAT GR8320x. Causal Inference. 3 pts.
Doctoral level introduction to casual inference, counterfactuals, causal estimates, ignorability, identifiability, and practical considerations. Matching, weighting, propensity scores, and sensitivily analyses.
STAT GR8325x and y. Topics In Advanced Statistics. 3 pts. Prerequisites: Statistics GR6108 and Department’s permission.
Available to SSP, SMP Topic to vary each time course is given. May be repeated for credit.
STAT GR8327x. Topics in Time Series Analysis. 3 pts
Time series models and efficient estimation of their parameters. Probabilistic and statistical aspects of non-linear and continuous-time models for time series data. GARCH, stochastic volatility, and nonlinear and non-Gaussian state-space models, continuous-time ARMA models that are both Brownaina- and Levy-driven, continuous-time GARCH (CO-GARCH), and stochastic volatility models. Recursive on-line identification of linear systems; adaptive filtering and control; Prediction theory and the Kalman filter; seasonal adjustment; frequency domain analysis.
STAT GR8329. Statistical Methods in Genetic Epidemiology. 3 pts. Prerequisites: Instructor’s permission.
Quick review of basic principles of genetics and genetic epidemiology; classical linkage analysis; genetic association studies; quantitative traits, haplotype relative risk and transmission disequilibrium methods, multipoint methods, methods for multigenic traits; genome-wide scans.
STAT GR8335. Statistical Methods in fMRI Research. 3 pts. Prerequisites:STAT W4105, W4107, W4315 or the equivalent
Introduction to mathematical and statistical aspects of functional magnetic resonance imaging; Data collection, image reconstruction, experimental design, pre-processing, model fitting, and inference. Extensive reading in classic and current literature.
Ph.D Working Groups
STAT GR7010x and y. Applied Probability and Risk Seminar.
A seminar on topics in Applied Probability and Risk.
STAT GR8990x and y. Research in Quantitative Political Science. 3 pts.
Research in quantitative methods in political science and in social science in general. Application area include American politics, comparative politics, social networks, law and public policy.