Statistics
24 Hillhouse, 432.0666
M.A., Ph.D.
Chair
Andrew Barron
Director of Graduate Studies
Marten Wegkamp (Rm 207, 24 Hillhouse, marten.wegkamp@yale.edu)
Professors
Donald Andrews (Economics), Andrew Barron, Joseph Chang, John Hartigan, Theodore Holford (Epidemiology & Public Health; Biostatistics), Peter Phillips (Economics), David Pollard
Associate Professors
Nicolas Hengartner, Junhyong Kim (Ecology & Evolutionary Biology), Marten Wegkamp, Heping Zhang (Epidemiology & Public Health; Biostatistics)
J. W. Gibbs Instructor
Dragan Radulovic
Fields of Study
Fields comprise the main areas of statistical theory (with emphasis on foundations, Bayes theory, decision theory, nonparametric statistics, probability theory (stochastic processes, asymptotics, weak convergence), information theory, econometrics, classification, statistical computing, and graphical methods.
Special Admissions Requirements
GRE scores for the General Test and for the Subject Test in the area of the undergraduate major should accompany an application. All applicants should have a strong mathematical background, including advanced calculus, linear algebra, elementary probability theory, and at least one course providing an introduction to mathematical statistics. An undergraduate major may be in statistics, mathematics, computer science, or in a subject in which significant statistical problems may arise. For those whose native language is not English, the Test of English as a Foreign Language (TOEFL) scores are required.
Special Requirements for the Ph.D. Degree
There is no foreign language requirement. Normally during the first two years, fourteen term courses in this and other departments are taken to prepare students for research and practice of statistics. These include courses devoted to case studies and practical work, for which students prepare a written report and give an oral presentation. The qualifying examination consists of three parts: a written report on an analysis of a data set, a written examination on theoretical statistics, and an oral examination. The examination is taken not later than when scheduled by the department in the middle of the second year, with provision for one subsequent reexamination of one or more parts in the event that a student does not pass the first time. All parts of the qualifying examination must be completed before the beginning of the third year. A prospectus for the dissertation should be submitted no later than the first week of March in the third year. The prospectus must be accepted by the department before the end of the third year if the student is to register for a fourth year. Upon successful completion of the qualifying examination and the prospectus (and meeting of Graduate School Requirements), the student is admitted to candidacy.
Master's Degree
M.A. (en route to the Ph.D.). This degree may be awarded upon completion of eight term courses and two terms of residence.
Master's Degree Program. Students are also admitted directly to a terminal master's degree program. To qualify for the M.A., the student must successfully complete eight term courses, chosen in consultation with the director of graduate studies. Full-time students must take a minimum of three courses per term. Part-time students are also accepted into the master's degree program.
Program materials are available upon request to the Director of Graduate Studies, Department of Statistics, Yale University, PO Box 208290, New Haven CT 06520-8290; e-mail, susan.jackson-mack@yale.edu.
Courses
STAT 501–506, Introduction to Statistics.
A basic introduction to statistics, including numerical and graphical summaries of data, probability, hypothesis testing, confidence intervals, and regression. Each course focuses on applications to a particular field of study and is taught jointly by two instructors, one specializing in statistics and the other in the relevant area of application.The Tuesday lecture, which introduces general concepts and methods of statistics, is attended by all students in STAT 501–506 together. The course separates for Thursday lectures (sections), which develop the concepts with examples and applications. Computers are used for data analysis.These courses are alternatives; they do not form a sequence and only one may be taken for credit.
STAT 501au, Introduction to Statistics: Life Sciences. Ashley Carter, Joseph Chang. TTh 1–2.15
Statistical and probabilistic analysis of biological problems presented with a unified foundation in basic statistical theory. The problems are drawn from genetics, ecology, epidemiology, and bioinformatics. Also E&EB 510au.
STAT 502au, Introduction to Statistics: Political Science. Rose Razaghian, Joseph Chang. TTh 1–2.15
Statistical analysis of politics and quantitative assessments of public policies. Problems presented with reference to a wide array of examples: public opinion, campaign finance, racially motivated crime, and health policy.
STAT 503au, Introduction to Statistics: Social Sciences. Donald Green, Joseph Chang. TTh 1–2.15
Introduction to probability and statistics with emphasis on experimental design and data analysis. Survey of many of the great experiments in social science. Topics include obedience to authority, conformity to social pressure, and susceptibility to perceptual distortions.
STAT 504au, Introduction to Statistics in Psychology. Thomas Brown, Joseph Chang. TTh 1–2.15
Statistical and probabilistic analysis of psychological problems presented with a unified foundation in basic statistical theory. The problems are drawn from studies of sensory processing and perceptions, development, learning, and psychopathology.
STAT 506au, Introduction to Statistics: Data Analysis. Mihaela Aslan, Joseph Chang. TTh 1–2.15
An introduction to probability and statistics, with emphasis on data analysis.
STAT 530bu, Introductory Data Analysis. David Pollard. MW 2.30–3.45
Survey of statistical methods: plots, transformations, regression, analysis of variance, clustering, principal components, contingency tables, and time series analysis. Uses SPLUS and Web data sources. After or concurrent with STAT 501a.
STAT 541au, Probability Theory. Marten Wegkamp. MWF 9.30–10.20
A first course in probability theory: probability spaces, random variables, expectations and probabilities, conditional probability, independence, some discrete and continuous distributions, central limit theorem, Markov chains, probabilistic modeling. After or concurrent with MATH 120a or b or the equivalent.
STAT 542bu, Theory of Statistics. Andrew Barron. MWF 9.30–10.20
Principles of statistical analysis: maximum likelihood, sampling distributions, estimation; confidence intervals; tests of significance; regression; analysis of variance; and the method of least squares. Some statistical computing. After STAT 541au and concurrently with or after MATH 222a or b or 225a or b or the equivalent.
STAT 551bu, Stochastic Processes. Dragan Radulovic. MW 1–2.15
Introduction to the study of random processes, including Markov chains, Markov random fields, martingales, random walks, Brownian motion, and diffusions. Techniques in probability such as coupling and large deviations. Applications to image reconstruction, Bayesian statistics, finance, probabilistic analysis of algorithms, genetics, and evolution. After STAT 541 or the equivalent.
STAT 600bu, Advanced Probability. David Pollard. TTh 2.30–3.45
Measure theoretic probability, conditioning, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems, martingales. Some knowledge of real analysis is assumed.
STAT 602b, Central Limit Theorem. Dragan Radulovic. HTBA
Central limit theorem (CLT) plays a key role in numerous statistical applications and has imbedded itself in many theoretical models. The proposed topics course covers (besides the historical accounts and the obvious "standard" CLT) the following topics: the "infinite variance case" (P-stable limits, infinite divisible laws, and Poisson [mu] limits), "dependence case" (alpha and beta mixing, CLT for time series and Markov chains), "multidimensional extension" (empirical processes, Banach space valued random variables) and the bootstrap for the above. Each of the above topics is motivated by real-life problems. Although no specific prerequisite courses are required, the knowledge of measure-theoretical probability (STAT 600) is strongly encouraged. Times to be arranged at organizational meeting.
STAT 610a, Statistical Inference. Andrew Barron.
A systematic development of the mathematical theory of statistical inference covering methods of estimation, hypothesis testing, and confidence intervals. An introduction to statistical decision theory. Undergraduate probability at the level of STAT 541a assumed.
STAT 612au, Linear Models. David Pollard. TTh 9–10.15
The geometry of least squares; distribution theory for normal errors; regression, analysis of variance, and designed experiments; numerical algorithms (with particular reference to S-plus); alternatives to least squares. Generalized linear models. Linear algebra and some acquaintance with statistics assumed.
STAT 618a, Asymptotic Theory. Marten Wegkamp. HTBA
A careful introduction to asymptotic methods in mathematical statistics. Topics include: consistency and asymptotic distributions, edgeworth expansions, M-estimators, contiguity, local asymptotic normality, efficiency, likelihood ratio theory, Le Cam's theory for convergence of experiments, bootstrap. After STAT 600b and STAT 610b. Times to be arranged at organizational meeting.
STAT 625a, Case Studies. Jonathan Reuning-Scherer.
Thorough study of some large data sets on such topics as second-hand smoking, crashes in small cars, reticulate evolution, bloc voting, and Connecticut educational standards.
STAT 626b, Practical Work. Staff.
Individual one-term projects, with students working on studies outside the department, under the guidance of a statistician.
STAT 645b, Statistical Methods in Genetics and Bioinformatics. Joseph Chang. HTBA
Stochastic modeling and statistical methods applied to problems such as mapping quantitative trait loci, analyzing gene expression data, sequence alignment, and reconstructing evolutionary trees. Statistical methods include maximum likelihood, Bayesian inference, Monte Carlo Markov chains, and some methods of classification and clustering. Models introduced include variance components, hidden Markov models, Bayesian networks, and coalescent. Recommended background: STAT 541, STAT 542. Prior knowledge of biology is not required. Times to be arranged at organizational meeting.
STAT 653b, Bayes Theory. John Hartigan. HTBA
Axioms and interpretations of probability. Construction of probability distributions. Optimality of Bayes procedures. Martingales. Asymptotics. Markov sampling. Robustness against violations in the assumed distributions. Choice among models. Times to be arranged at organizational meeting.
STAT 660b, Multivariate Statistical Analysis in the Environmental Sciences. Jonathan Reuning-Scherer. HTBA
An introduction to the analysis of multivariate data. Topics to include multivariate analysis of variance (MANOVA), principal components analysis, cluster analysis (hierarchical clustering, k-means), canonical correlation, multidimensional scaling, and factor analysis. Some analysis of multivariate spatial data may be included. Emphasis is placed on practical application of multivariate techniques to a variety of natural and social examples in the environmental sciences. Students are required to select a dataset early in the term for use throughout the term. There are regular assignments and a final project. Times to be arranged at organizational meeting. Also F&ES 844b.
STAT 661bu, Data Analysis. John Hartigan. MW 2.30–3.45
By analyzing data sets using the S-plus statistical computing language, a selection of statistical topics are studied: linear and nonlinear models, maximum likelihood, resampling methods, curve estimation, model selection, classification, and clustering. Weekly sessions are held in the Social Sciences Statistical Laboratory. After STAT 542 and MATH 222 or 225 or the equivalents.
STAT 664bu, Information Theory. Edmund Yeh. TTh 9–10.15
Foundations of information theory in communications, statistical inference, statistical mechanics, probability, and algorithm complexity. Quantities of information and their properties: entropy, conditional entropy, divergence, mutual information, channel capacity. Basic theorems of data compression and coding for noisy channels. Applications in statistics, communication networks, and finance. After STAT 541a.
STAT 665bu, Introduction to Function Estimation. Marten Wegkamp. MW 11.30–12.45
A practical introduction to curve estimation techniques, such as nearest neighbors, regression splines, series estimators, local regression smoothers, and neural networks, with discussion of boundary effects, model and bandwidth selection, goodness of fit, and confidence intervals/bands. Further topics include bootstrap, dimension reduction, boosting, pattern recognition, and density estimation.
STAT 674au, Analysis of Spatial and Time Series Data. Dragan Radulovic. TTh 1–2.15
Study of statistical models that are useful for describing data collected over space or time. Models include frequency domain and time domain analysis of time series; state space models and Kalman filters; point processes; Gibbs processes and random fields.
STAT 676a, Some Topics in Portfolio Selection. Andrew Barron. HTBA
A study of distributional properties of compounded wealth in repeated gambling and in stock market investment. Wealth concentration inequalities. Strategies of highest concentrated wealth. Normal theory for log-wealth. Relationship to maximum likelihood theory in statistics and to the asymptotic equipartition property in physics and information theory. Greedy strategies. Universal portfolios and their relationship to Bayes methodology. The ratio of idealized wealth (best with hindsight) to actual wealth and the properties of this ratio, both for stochastic stock price sequences and its minimax behavior for arbitrary price sequences. Fast algorithms for universal portfolios. Times to be arranged at organizational meeting.
STAT 695a, Internship in Statistical Research. Marten Wegkamp.
The internship is designed to give students an opportunity to gain practical exposure to problems in the analysis of statistical data, as part of a research group within industries such as: medical and pharmaceutical research, finance, information technologies, telecommunications, public policy, and others. The internship experience often serves as a basis for the Ph.D. dissertation. Students work with the director of graduate studies and other faculty advisers to select suitable placements. Students submit a one-page description of their internship plans to the DGS by May 1, which will be evaluated by the DGS and other faculty advisers by May 15. Upon completion of the internship, students submit a written report of their work to the DGS, no later than October 1. The Internship is graded on a Satisfactory/Unsatisfactory basis, and is based on the student's written report and an oral presentation. This course is an elective requirement for the Ph.D. degree. Prerequisites: completion of one semester of the Ph.D. program.
STAT 700, Departmental Seminar.
Important activity for all members of the department. See weekly seminar announcements.
Next: The Whitney Seminars
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