Statistics

24 Hillhouse, 432.0666
M.A., Ph.D.

Chair
Andrew Barron

Director of Graduate Studies
Nicolas Hengartner (Rm 207, 24 Hillhouse, nicolas.hengartner@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), Heping Zhang (Epidemiology & Public Health; Biostatistics)

Assistant Professor
Marten Wegkamp

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 Degrees
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. Joseph Chang, Junhyong Kim.
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.

STAT 502au, Introduction to Statistics: Political Science. Joseph Chang.
Statistical analysis of social science problems, primarily drawn from political science and sociology, presented with a unified foundation in basic statistical theory.

STAT 503au, Introduction to Statistics: Sociology. Joseph Chang, Eric Kostello. Tues/Thurs 1-2.15
An introduction to probability and statistics, with emphasis on applications to sociology. Also SOCY 580au.

STAT 504au, Introduction to Statistics in Psychology. Joseph Chang, Thomas Brown.
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. Also NSCI 540a.

STAT 505au, Introduction to Statistics: Environmental Sciences. Joseph Chang, Jonathan Reuning-Scherer.

An introduction to probability and statistics with emphasis on applications to forestry and environmental sciences, presented with a unified foundation in basic statistical theory.

STAT 506au, Introduction to Statistics: Data Analysis. Joseph Chang, Nicolas Hengartner. Tues/Thurs 1-2.15
An introduction to probability and statistics, with emphasis on data analysis.

STAT 530bu, Introductory Data Analysis. John Hartigan.
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.
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.
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 concurrent with or after MATH 222b or 225a or b or the equivalent.

STAT 551bu, Stochastic Processes. David Pollard.
A study of random processes, including Markov chains, Markov random fields, martingales, random walks, Brownian motion, and diffusions. Introduction to certain modern techniques in probability like 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. Marten Wegkamp.
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 603a, Stochastic Calculus. David Pollard.
Martingales in discrete and continuous time, Brownian motion, sample path properties, predictable processes, stochastic integrals with respect to Brownian motion and semimartingales, stochastic differential equations. Applications mostly to counting processes and finance. Prerequisite: knowledge of measure-theoretic probability at the level of STAT 600, although some key concepts, such as conditioning, are reviewed. After STAT 600.

STAT 610a, Statistical Inference. David Pollard.
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. Marten Wegkamp.
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 625a, Case Studies. John Hartigan.
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 645a, Topics in the Statistical Analysis of Genomic Data. Joseph Chang.
Several recently developed statistical methods either have already played an important role in the analysis of genomic and post-genomic data or appear to be promising candidates to do so. We study hidden Markov models, Bayesian networks, support vector machines and kernel methods, and perhaps other topics to be determined. For each topic, instructors present introductory lectures on the statistical theory, models, and methods of analysis. Students work on projects and present results, which may include computer implementations of the statistical techniques, analyses of biological sequence and gene expression data using available programs, and reports on research papers. Although there are no specific prerequisites, the course makes substantial use of probability theory, statistics, introductory biology, and computation; students without background in some of these areas may need to do additional work and should consult the instructors before enrolling.

STAT 661bu, Data Analysis. Nicolas Hengartner.
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. Andrew Barron.
Foundations of information theory in mathematical 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 channel coding. Applications in statistics and finance. After STAT 541a.

STAT 665bu, Introduction to Function Estimation. Nicolas Hengartner.
A practical introduction to modern curve estimation techniques, such as nonlinear regression, 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 estimation under shape restriction, pattern recognition, inverse problems, hazard estimation, and density estimation.

STAT 674au, Analysis of Spatial and Time Series Data. John Hartigan. Tues/Thurs 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 700, Departmental Seminar.
Important activity for all members of the department. See weekly seminar announcements.

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