Course Descriptions

Graduate Courses in the Department of Statistics
Courses numbered between 36-625 and 36-740 are at the Master's level, those between 36-751 and 36-799 represent Ph.D.-level core courses, and those above 36-800 are Ph.D.-level elective courses. The courses with 46- prefixes are part of the Master of Science in Computational Finance program.

36-625 Probability and Mathematical Statistics I 12 units
This course is a rigorous introduction to the mathematical theory of probability, and it provides the necessary background for the study of mathematical statistics and probability modeling. A good working knowledge of calculus is required. Topics include combinatorial analysis, conditional probability, generating functions, sampling distributions, law of large numbers, and the central limit theorem. Students studying Computer Science, or considering graduate work in Statistics or Operations Research, should carefully consider taking this course instead of 36-225 after consultation with their advisor. Not open to students who have received credit for 36-217 or 36-225. Prerequisite: 21-122 and 21-241 and (21-256 or 21-259).

36-626 Probability and Mathematical Statistics II 12 units
An introduction to the mathematical theory of statistical inference. Topics include likelihood functions, estimation, confidence intervals, hypothesis testing, Bayesian inference, regression, and the analysis of variance. Not open to students who have received credit for 36-226. Students studying Computer Science should carefully consider taking this course instead of 36-220 or 36-226 after consultation with their advisor. Prerequisite: 36-625.

36-701 Perspectives on Statistics 3 units
Students are introduced to the faculty and their interests, the field of statistics, and the facilities at Carnegie Mellon. Each faculty member gives at least one elementary lecture on some topic of his or her choice. In the past, topics have included: the field of statistics and its history, large-scale sample surveys, survival analysis, subjective probability, time series, robustness, multivariate analysis, psychiatric statistics, experimental design, consulting, decision-making, probability models, statistics and the law, and comparative inference. Students are also given information about the libraries at Carnegie Mellon and current bibliographic tools. In addition, students are instructed in the use of the Departmental and University computational facilities and available statistical program packages.

36-703 Intermediate Probability 12 units
The basic definitions of probability and topics such as conditional probability and conditional expectation are reviewed. Discrete-time Markov chains and the Poisson process are introduced and discussed in detail. Additional topics covered include birth and death processes, continuous-time Markov chains, and elementary renewal processes. Material is presented at the level of Taylor and Karlin, An Introduction to Stochastic Modeling, 3rd ed.

36-705 Intermediate Statistics 12 units
Some elementary concepts of statistics and probability are reviewed, including convergence concepts. Then the concepts of sufficiency, likelihood, and information are introduced. Several methods of estimation, such as maximum likelihood estimation and Bayes estimation, are studied, and some approaches to comparing different estimation procedures are discussed. The class also covers tests of hypotheses, and for both estimation and testing, methods for finding and characterizing optimal decisions are considered. Additional topics are covered depending on time and interest. Material is presented at the level of Bickel and Doksum, Mathematical Statistics, and Casella and Berger, Statistical Inference.

36-707 Regression Analysis 12 units
This is a course in data analysis using mutiple linear regression. Topics covered include simple linear regression, ordinary least squares and weighted least squares, the geometry of least squares, quadratic forms, F tests and ANOVA tables, residuals, outlier detection, and identification of influential observations, variable selection methods, and modern regression techniques. Essential background in linear algebra is reviewed where necessary. When time permits other topics such as nonlinear regression and robust estimation will be discussed. Practice in data analysis is obtained through course projects.

36-708 Linear Models and Experimental Design 12 units

This is a course in experimental design and the analysis of experimental data using linear models. The first part of the course covers theoretical background: the multivariate normal distribution, distribution of quadratic forms, principles of analysis of variance, and the Bayesian approach to linear models. The second part concerns basic experimental designs: latin squares, factorial designs, fractional factorials, and nested designs. In the final component of the course, random effects and mixed models, repeated measures, and longitudinal analysis are discussed.

36-711 Statistical Computing 12 units
This course introduces students to a range of computational techniques that are important to statistics. The topics covered include numerical linear algebra, numerical optimization, graphical techniques, numerical approximations, numerical integration and Monte Carlo methods. Use of statistical packages (Splus, SAS) and programming libraries (IMSL) is also illustrated.

36-712 Statistical Approaches to Learning and Discovery (cross-listed from 10-602) 12 units
This course builds on the material presented in 10-601 (Machine Learning), introducing new learning methods and going more deeply into their statistical foundations and computational aspects. Topics include: approaches to statistical inference; overview of regression, classification, and clustering; the EM algorithm for exponential families; data augmentation and Markov Chain Monte Carlo algorithms; techniques for supervised and unsupervised learning. Applications and case studies from statistics and computer science are used to illustrate each topic.

36-713 Nonparametric Methods 6 units
This course introduces modern methods for nonparametric statistical inference. Topics include: statistical functionals, influence functions, the bootstrap, the jackknife, histograms, kernel density estimation, nonparametric regression, smoothing by orthogonal expansions, minimax function estimation, and wavelets. This course requires background knowledge at the level of 36-705.

36-720 Discrete Multivariate Analysis 6 units
Theory of maximum likelihood in exponential families developed in Intermediate Statistics, 36-705, is applied to the analysis of binary and categorical data. Logistic regression and loglinear models are introduced, and students gain experience in using statistical program packages and interpreting results.

36-722 Continuous Multivariate Analysis 6 units
This course covers the multivariate normal distributions and its marginal, conditional and sampling distributions. Other topics include Hotelling's T-squared, MANOVA, discriminant analysis, and principal components. The data analytic portions of the class use standard statistical packages.

36-724 Applied Bayesian Methods 6 units
This course is an introduction to practical Bayesian methodology. The use of conjugate families, introduced in Intermediate Statistics, 36-705, is discussed. Building on techniques in Statistical Computing, 36-711, methods for calculating posterior distributions are presented, as is the concept of hierarchical model. The emphasis throughout is on the application of Bayesian thinking to problems in data analysis.

36-726 Statistical Practice 6 units
Students are taught how to structure a consulting session, elicit and diagnose a problem, manage a project, and report an analysis. The class will participate in meetings with industrial and academic clients.

36-728 Time Series Analysis 6 units
Basic concepts of linear time series are presented, including stationarity, causality, invertibility, autoregressive moving average models, and forecasting. Methods for building AR, MA and ARMA models are discussed. The course also introduces ARIMA models, and briefly touches on state space models and the Kalman filter. Real-world data are used as illustrative examples.

36-743 Statistical Methods for the Behavioral and Social Sciences 12 units
This course offers a one-semester overview of statistical methods for the analysis of data obtained from experiments and observational studies. The course is aimed at graduate students who are preparing to design, implement, analyze, and report their research findings. The target audience is behavioral and social scientists. The students must have some knowledge of basic statistical concepts such as means, standard deviations, histograms, the normal and t-distributions, but they need not be familiar with calculus or matrix algebra. Topics include linear regression, analysis of variance, logistic regression, model checking and refinement, strategies for variable selection, repeated measures, and exploratory tools for summarizing multivariate responses.

36-746 Statistical Methods for Neuroscience 12 units
This course provides a brief survey of statistical methods that are of use in cognitive neuroscience. The first part of the course will present a compressed version of material often covered in a semester-long course in elementary statistics. The latter part of the course will introduce various more advanced methods. Topics include Probability (laws of probability, conditional probability, Bayes' Theorem, random variables, Binomial, Poisson, and Normal distributions, and Poisson and other point processes), Exploratory Data Analysis (Descriptive methods for single samples and multiple samples, scatterplot smooths, histograms, and density estimators), Elementary Statistical Inference (standard errors and confidence intervals, goodness-of-fit and significance tests, ANOVA and regression, and maximum likelihood and Bayesian inference). Additional topics may include Bayesian classification, ROC curves, Information theory, Fourier analysis and signal processing, Multivariate analysis, PCA and ICA, the Bootstrap, nonparametric regression, and integrate-and-fire models.

36-747/36-247 Statistics for Laboratory Science 9 units
This course is a single-semester comprehensive introduction to statistical analysis of data for students in biology and chemistry. Topics include exploratory data analysis, elements of computer programming for statistics, basic concepts of probability, statistical inference, and curve fitting. In addition to two lectures, students attend a computer lab each week. Not open to students who have received credit for 36-202, 36-208/70-208, 36-220, 36-226 or 36-326. Prerequisite: 21-111 or 21-116 or 21-121.

36-749/36-309 Experimental Design for Behavioral and Social Sciences 9 units
Statistical aspects of the design and analysis of planned experiments are studied in this course. A clear statement of the experimental factors will be emphasized. The design aspect will concentrate on choice of models, sample size and order of experimentation. The analysis phase will cover data collection and computation, especially analysis of variance, and will stress the interpretation of results. In addition to weekly lecture, students will attend a computer lab once a week. Prerequisite: 36-202, 36-220, or 36-247.

36-752 Advanced Probability Overview 12 units
This course is a one-semester overview of topics in Probability Theory. After a brief introduction to measure and integration theory, the focus will be on issues of immediate use to statisticians, such as modes of convergence, limit theorems, laws of large numbers, martingales, and other topics as time allows.

36-754 Probability Theory and Stochastic Processes 12 Units
This course introduces advanced topics in Probability theory such as Brownian motion, Markov processes, stationary processes, stochastic integration, etc.

36-755 Advanced Statistical Theory I 12 units per semester
36-756 Advanced Statistical Theory II
This course involves intensive study of the fundamental topics in statistical theory: sufficient statistics, estimation, hypothesis testing, exchangeability, invariance, posterior distributions, decision theory, large sample theory, and optimality criteria. Open to students in the statistics Ph.D. or joint degree programs only.

36-757 Advanced Data Analysis I 12 units per semester
36-758 Advanced Data Analysis II
Advanced Data Analysis (ADA) is a Ph.D. level seminar on advanced methods in statistics, including computationally intensive smoothing, classification, variable selection and simulation techniques. During 36-757, you work with the seminar instructor to identify an ADA project for yourself. The ADA project is an extended project in applied statistics, done in collaboration with an investigator from outside the Department, under the guidance of a faculty committee, culminating in a publishable paper that is presented orally and in writing in 36-758.

36-784 Parallel Computing in Statistics Mini A4
We investigate the theory and application of parallel algorithms in common statistical problems. Students are required to attend lectures, lead group discussions and complete a course project involving an implementation of a non-trivial parallel method to a statistical application. Topics addressed include basic parallel paradigms (SIMD, MISD and MIMD), parallelized algebraic methods, Monte Carlo methods with parallel constructions, applications of multiprocessor and GPU computing, and other topics of interest.

36-911 Seminar in Foundations of Statistics 12 units
This seminar offers an opportunity to read and discuss "classic" texts and to investigate their impact on current statistical practice. Topics vary from year to year. Three recent selections were these: A. Wald's contribuitons to statistical decision theory, with an emphasis on his use of minimax and sequential decision rules, for which the primary text was Statistical Decision Functions; L.F. Savage's theory of personal probability for which the primary text was The Foundations of Statistics; R.A. Fisher's account of varieties of statistical inference, with an emphasis on fiducial probability, for which the primary text was Statistical Methods and Scientific Inference.

36-912 Seminar in Psychiatric Statistics 12 units

36-995 Reading and Research units to be assigned

36-997 Practicum in Statistics

36-998 Qualifying Examination for the Degree of Doctor of Philosophy

36-999 Final Public Oral Examination for the Degree of Doctor of Philosophy

46-921 Introduction to Probability 6 units
The objective of this course is to introduce the basic ideas and methods of calculus-based probability theory and to provide a solid foundation for other MSCF courses based on probability theory. Topics include basic results on probability and conditional probability, random variables and their distribution, expected values, moment generating functions transformations of random variables and vectors, simulation, laws of large numbers and the central limit theorem. Representative text is Probability and Statistics, by Morris DeGroot and Mark Schervish, Third Edition, 2002.

46-923 Introduction to Statistical Inference

The objective of this course is to introduce the basic ideas and methods of statistical inference and the practice of statistics, especially estimation and basic regression analysis. The statistical package S-PLUS will be introduced. This package is used throughout the MSCF curriculum. Mathematical statistical theory will be supplemented by simulation and data analysis methods to illustrate the theory. This course will provide a solid foundation for subsequent MSCF courses in statistics. Representative text is Probability and Statistics, by Morris DeGroot and Mark Schervish, Third Edition, 2002.

46-926 Linear Financial Models/Equity Portfolio Management 6 units
This is a course in regression analysis and linear models with application to equity portfolio management. Basic methods taught in the course include simple and multiple linear regression, model selection, residual analysis, diagnostics, detection of multi-collinearity, nonstandard conditions and transformations. Principal components and factor analysis are also introduced. Examples will be taken from financial models, including the CAPM and multi-factor with applications to portfolio selection and asset allocation. Representative texts are: Campbell, J.Y., Lo, A.W. and MacKinlay, A.C. (1997). The Econometrics of Financial Markets. Princeton University Press; Modern Applied Statistics with Splus, by Venables and Ripley, Third Edition Springer-Verlag.

46-929 Financial Time Series Analysis
This course introduces time series methodology to the MSCF program. Emphasis will be placed on the data analytic aspects related to financial applications. Both the time domain and the frequency domain approach will be introduced. Topics studied in this course include univariate ARIMA modeling, forecasting, seasonality, model identification and diagnostics. Related applications to finance such as time series modeling of equity returns, trading day effects, and volatility estimations will also be discussed. In addition, recent advancements in financial time series including the unit root phenomenon, cointegration, GARCH and stochastic volatility modeling, trend break analysis and nonlinearity will be covered. Representative text are: Chatfield, C (1996). The Analysis of Time Series, (5th Ed.) Chapman and Hall, New York. Mills, T.C. (1993). The Econometric Modeling of Financial Time Series, Cambridge University Press, Cambridge. Campbell, J.Y., Lo, A.W. and MacKinlay, A.C.(1997). The Econometrics of Financial Markets. Princeton University Press.

46-932 Simulation Methods for Option Pricing 6 units
This course initially presents standard topics in simulation including random variable generation, variance reduction methods and statistical analysis of simulation output. The course then reviews papers from the current finance literature to illustrate the application of these methods to derivative security pricing. The topics addressed include importance sampling, martingale control variables, stratification and the estimation of derivatives. Additional topics include the use of low discrepancy sequences (quasi-random numbers), pricing American options and scenario simulation for risk management.

46-936 Statistical Arbitrage
This course intends to provide students with concepts and techniques for statistically and econometrically based trading. The course begins with the general principles of arbitrage pricing theory and the statistical nature of the price and volatility fluctuations in financial markets. It introduces the ideas of market neutral strategies, and provides the statistical techniques required for identifying and exploiting pricing inefficiencies. Various statistical strategies will be covered, including pairs trading, cointegration-based trading, data mining, as well as strategies using the information from derivatives markets. We will demonstrate how to search for arbitrage strategies based on intra-day patterns, long-term patterns, multi-equity relationships. At the end we stress that statistical arbitrage is not riskless, and we discuss how to assess the risk, arising from model misspecification and inappropriate estimation. The topics covered are particularly relevant for proprietary trading, such as in the context of hedge funds.