### 500-Level Statistics Courses (500-535)

### Statistics 500: Applied Statistics I

Linear models; definitions, fitting, identifiability, collinearity, Gauss-Markov theorem, variable selection, transformation, diagnostics, outliers and influential observations. ANOVA and ANCOVA. Common designs. Applications and real data analysis are stressed,with students using the computer to perform statistical analyses.

* Pre-requisite: *MATH 217 or 417 and STATS 250/350, 280, or 426.

### Statistics 501: Applied Statistics II

Generalized linear models including logistics regression, Poisson regression, contingency tables. Random effects and repeated measures. Modern regression techniques. Regression and classification trees. Neural networks.

* Pre-requisite: *STATS 500.

### Statistics 503: Applied Multivariate Analysis

Topics in applied multivariate analysis including Hotelling's T-squared, multivariate ANOVA, discriminant functions, factor analysis, principal components, canonical correlations, and cluster analysis. Selected topics from: Maximum likelihood and Bayesian methods, robust estimation, and survey sampling. Applications and data analysis using the computer is stressed.

* Pre-requisite: *STATS 500.

### Statistics 504: Seminar on Statistical Consulting

Introduces students to key aspects of statistical consulting and data analysis activities: Problem solving and real applications, data analysis, reports, and presentations.

* Advisory Pre-requisite: *MATH 417, and STATS 250/350, 280, or 426.

### Statistics 505: Econometric Analysis I (ECON 671)

Econ 671 and 672 form the basic required sequence in econometrics for all doctoral students. Their purpose is to provide Ph.D. students with the training needed to do the basic quantitative analysis generally understood to be part of the background of all modern economists. This includes: the theory and practice of testing hypotheses, statistical estimation theory, the basic statistical theory underlying the linear model, an introduction to econometric methods, and the nature of the difficulties which arise in applying statistical procedures to economic research problems.

*Permission of instructor required to register.*

### Statistics 508: Statistical Analysis of Financial Data

This course will cover basic topics involved in modeling and analysis of financial data. These include linear and non-linear regression, nonparametric and semi-parametric regression, selected topics on the analysis of multivariate data and dimension-reduction, and time series analysis. Examples and data from financial applications will be used to motivate and illustrate the methods.

*This course is restricted to Financial Engineering students only.*

### Statistics 509: Statistics for Financial Data

### Statistics 510: Mathematical Statistics

This course introduces the essential concepts of probability and emphasizes topics that are important for statistical theory. Topics include counting, probability and conditional probability, independence, random variables, distribution functions, modeling dependence, transformations, quantiles, order statistics, laws of large numbers, central limit theorem, and sampling distributions.

* Prereq: *MATH 215 or equivalent.

### Statistics 511: Mathematical Statistics II

This is an introductory course to statistical estimation and inference. It is designed to give students sufficient mathematical statistics background so that they can utilize the theory in practice. The course will cover the following topics: Univariate and multivariate families of distributions, sampling distributions and relevant properties, likelihood principles, point estimation, inference procedures, large sample properties, and selected topics in contemporary statistical methods.

* Pre-requisite: *STATS 425, MATH 425, OR STATS 510 OR equivalent courses in probability.

### Statistics 520: Mathematical Methods in Statistics

This course provides the mathematical background for theoretical Ph.D.-level courses in statistics and probability. The course reviews basic notions from matrix algebra and real analysis. It then introduces students to measure theory and integration. In particular, the content covers definition of measures and measurable functions, convergence theorems, Lebesgue integration, Lp spaces, signed measures, Radon-Nikodym theorem, and integration on product spaces.

* Pre-requisite: *MATH 451 or equivalent course in real analysis.

### Statistics 525: Probability Theory (MATH 525)

This course covers axiomatic probability; combinatorics; random variables and their distributions; special distributions; joint, marginal and conditional distributions; expectation; the mean, variance, and moment generating function; induced distributions; sums of independent random variables; the law of large numbers; the central limit theorem. Optional topics drawn from: random walks, Markov chains, and/or martingales.

* Advisory Pre-requisite:* MATH 451 (strongly recommended). STATS 425 would be helpful. Students with credit for MATH 425/STATS 425 can elect MATH 525/STATS 525 for only one credit.

### Statistics 526: Discrete State Stochastic Processes (MATH 526)

Review of discrete distributions; generating functions; compound distributions, renewal theorem; modeling of systems as Markov chains; first properties; Chapman-Kolmogorov equations; return and first passage times; classification of states and periodicity; absorption probabilities and the forward equation; stationary distributions and the backward equation; ergodicity; limit properties; application to branching and queueing processes; examples from engineering, biological and social sciences; Markov chains in continuous time; embedded chains; the M/G/1 queue; Markovian decision processes; application to inventory problems; other topic at instructor's discretion.

* Pre-requisite: *STATS 525 or EECS 501

### Statistics 531: Analysis of Time Series (ECON 677)

Decomposition of series; trends and regression as a special case of time series; cyclic components; smoothing techniques; the variate difference method; representations including spectrogram, periodogram, etc.; stochastic difference equations autoregressive schemes, moving averages; large sample inference and prediction; covariance structure and spectral densities; hypothesis testing and estimation and applications and other topics.

* Pre-requisite: *STATS 426

### Statistics 535: Reliability (IOE 562)

This course covers the important reliability concepts and methodology that arise in modeling, assessing, and improving product reliability and in analyzing field and warranty data. Topics are selected from the following: Basic reliability concepts, common parametric models for component reliability, censoring schemes, analysis of time-to-failure data, accelerated testing for reliability assessment, modeling and analyzing repairable systems reliability, analysis of warranty and field-failure data, maintenance policies and availability, reliability improvement through experimentation.

* Pre-requisite:* STATS 425 and 426 (or IOE 316 and 366)

### 500-Level Statistics Courses (545-580)

### Statistics 545: Data Analysis in Molecular Biology (BIOSTAT 646, BIOINFORMATICS 545)

The course will cover statistical methods used to analyze data in experimental molecular biology, with an emphasis on gene and protein expression array data. Topics: Data acquisition; databases; low level processing; normalization; quality control; statistical inference (group comparisons, cyclicity, survival); multiple comparisons; statistical learning algorithms; clustering; visualization; and case studies.

* Pre-requisite: *Graduate standing and STATS 400 (or equivalent) or permission of instructor.

* Advisory Pre-requisite: *Students should have a strong preparation in either biology or some branch of quantitative analysis (mathematics, statistics, or computer science), but not necessarily in both domains.

### Statistics 547: Probabilistic Modeling in Bioinformatics (Math 547)

Probabilistic models of proteins and nucleic acids. Analysis of DNA/RNA and protein sequence data. Algorithms for sequence alignment, statistical analysis of similarity scores, hidden Markov models, neural networks training, gene finding, protein family profiles, multiple sequence alignment, sequence comparison and structure prediction. Analysis of expression array data.

* Pre-requisite: *STATS 425 or BIOL 427 or BIOL CHEM 415; basic programming skills desirable. Graduate standing and permission of instructor.

### Statistics 548: Computations in Probabilistic Modeling in Bioinformatics (MATH 548)

This will be a computational laboratory course designed in parallel with Math/Stat 547. Weekly hands-on problems will be presented on the algorithms presented in the course, the use of public sequence databases, the design of hidden Markov models. Concrete examples of homology, gene finding, structure analysis.

* Pre-requisite: *STATS 425 or BIOL 427 or BIOL CHEM 415; basic programming skills desirable. Graduate standing and permission of instructor.

### Statistics 550: Bayesian Decision Analysis (IOE 560)

Axiomatic foundations for personal probability and utility; interpretation and assessment of personal probability and utility; formulation of Bayesian decision problems; risk functions, admissibility; likelihood principle and properties of likelihood functions; natural conjugate prior distributions; improper and finitely additive prior distributions; examples of posterior distributions, including the general regression model and contingency tables; Bayesian credible intervals and hypothesis tests; applications to a variety of decision-making situations.

* Pre-requisite: *STATS 425

### Statistics 553: Conceptual Foundations of Statistical Inference (PHIL 553)

This course will focus on conceptual issues in the foundations of probability theory and statistics. It is intended for graduate students with modest prior background in statistics. Probability theory will be reviewed, and elementary statistical techniques will be discussed. Course will evaluate the main philosophical interpretations of the probability calculus and resulting paradigms of statistical inference.

* Pre-requisite:* A course in statistical theory

*(e.g.*PSYCH 613, ECON 405) and graduate or advanced undergraduate standing, or permission of instructor.

### Statistics 560: Introduction to Nonparametric Statistics (BIOS 685)

Confidence intervals and tests for quantiles, tolerance regions, and coverages; estimation by U statistics and linear combination or order statistics; large sample theory for U statistics and order statistics; the sample distribution and its uses including goodness-of-fit tests; rank and permutation tests for several hypotheses including a discussion of locally most powerful rank and permutation tests; and large sample and asymptotic efficiency for selected tests.

* Pre-requisite:* STATS 426

### Statistics 570: Design of Experiments (IOE 570)

Basic topics and ideas in the design of experiments: randomization and randomization tests; the validity and analysis of randomized experiments; randomized blocks; Latin and Graeco-Latin squares; plot techniques; factorial experiments; the use of confounding and response surface methodology; weighing designs, lattice and incomplete block and partially balanced in complete block designs.

* Pre-requisite: *STATS 500 or background in regression. Graduate standing.

### Statistics 580: Methods and Theory of Sample Design (SOC 717/BIOS 617)

Theory underlying sample designs and estimation procedures commonly used in survey practice. Simple random sampling, stratification systematic sampling, cluster sampling, multistage sampling, sampling with probability proportional to size, replicated sampling, multiphase sampling. Post-stratification, ratio, regression and difference estimation. Variance estimation with complex sample designs: Taylor series method, repeated replications, jackknife repeated replications. Nonresponse weighting adjustments and imputation.

* Pre-requisite: *Three or more courses in Statistics and preferably a course in methods of survey sampling.

### 600-Level Statistics Courses

### Statistics 600: Linear Models

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.

* Prereq: *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 601: Analysis of Multivariate and Categorical Data

This is an advanced introduction to the analysis of multivariate and categorical data. Topics include: (1) dimension reduction techniques, including principal component analysis, multidimensional scaling and extensions; (2) classification, starting with a conceptual framework developed from cost functions, Bayes classifiers, and issues of over-fitting and generalization, and continuing with a discussion of specific classification methods, including LDA, QDA, and KNN; (3) discrete data analysis, including estimation and testing for log-linear models and contingency tables; (4) large-scale multiple hypothesis testing, including Bonferroni, Westphal-Young and related approaches, and false discovery rates; (5) shrinkage and regularization, including ridge regression, principal component regression, partial least squares, and the lasso; (6) clustering methods, including hierarchical methods, partitioning methods, K-means, and model-based clustering.

* Prereq: *STATS 600

### Statistics 604: Statistical Investigations in a Consulting Framework

Statistical Investigations in a Consulting Framework --- This course provides graduate students with a variety of modeling and data analysis opportunities through consultations with other researchers. The modeling activities include problems arising in the science, engineering and the humanities and will include both quantitative and qualitative features. The quantitative data will be modeled in various ways including mixed models, and as functional data. It is anticipated that each student will have some exposure to geographic information systems, survey data and qualitative data too and will examine alternative modeling approaches to those seen in a more conventional setting. Each student will learn to formulate alternative modeling approaches. A seminar will allow students and instructor to learn to formulate alternative modeling approaches. A seminar will allow students and instructor to learn the process of question formulation, and alternative strategies for analysis. Communication of findings will be a critical element of the course and evaluation of the learning.

### Statistics 605: Advanced Topics in Modeling and Data Analysis

This course covers recent developments in statistical modeling and data analysis. Topics include: (1) classification and machine learning, including support vector machines, recursive partitioning, and ensemble methods; (2) methods for analyzing sets of curves, surfaces and images, including functional data analysis, wavelets, independent component analysis, and random field models; (3) modern regression, including splines and generalized additive models, (4) methods for analyzing structured dependent data, including mixed effects models, hierarchical models, graphical models, and Bayesian networks; and (5) clustering, detection, and dimension reduction methods, including manifold learning, spectral clustering, and bump hunting.

* Prereq:* STATS 601

### Statistics 606: Statistical Computing

This course is an advanced introduction to numerical techniques used in statistics, modeling, and data analysis. Course topics include: background in numerical analysis (concepts of convergence rate and computational complexity), classical numerical techniques (root finding, quadrature), combinatorial enumerations, numerical linear algebra (linear solvers, matrix factorizations, eigenvalue problems), optimization (Newton-Raphson, gradient optimization, EM algorithm, constrained optimization, stochastic optimization), and simulation (pseudo-random numbers, importance and rejection sampling, Markov chain methods). The course will cover both theory and implementation. Statistical applications will include frequentist and Bayesian model estimation and inference for complex models including hierarchical linear and nonlinear models, random effects and latent variable models, signal and image processing methods, clustering and classification methods, and visualization.

* Prereq:* Some experience in computer programming is required. 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: Programming and Numerical Methods in Statistics

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.

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.

* Advisory Pre-requisites:* STATS 425, STATS 426. Computer Programming experience recommended. Department Consent Required.

### Statistics 608: Methods in Optimization Statistics

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.

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.

### Statistics 610: Statistical Inference

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.

* Pre-requisites: *MATH 451, STATS 425, and STATS 426 or equivalent courses in probability, statistics and real analysis.

### Statistics 611: Large Sample Theory

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.

* Pre-requisites:* STATS 610

### Statistics 612: Advanced Topics in Theoretical Statistics

This course deals with selected topics in theoretical statistics at an advanced level. Topics include stochastic convergence in metric spaces, empirical processes including the empirical distribution function, functional delta method and applications including large sample theory for nonparametric density estimation, and the large sample theory for bootstrap methods. Other possible topics include semi-parametric models and efficiency issues.

* Pre-requisites: *STATS 520 or equivalent course in measure theory and STATS 611

### Statistics 617: Advanced Topics in Quantitative Methodology

This course explores and critiques advanced methods for conducting quantitative research in the social sciences. A special topic is chosen for a particular semester, with relevant methods drawn from a wide variety of disciplines, including economics, education, epidemiology, psychology, sociology, and statistics. Particular attention is paid to quasi-experimental and observational research design.

* Pre-requisites: *Graduate level courses in Statistics at the level of 500 and 501 or permission of instructor.

### Statistics 620: Applied Probability and Stochastic Modeling

This course is an introduction to stochastic models that capture the evolution in time of various random phenomena and/or dynamical systems. Such phenomena/systems arise extensively in diverse areas of research, ranging from biology, to data networks and production planning. Emphasis is placed on modeling aspects as well as development of the underlying theory. Topics covered include: Markov chains in discrete and continuous time, Poisson and renewal processes, Brownian motion. Applications of these models in key scientific and engineering areas, such as genetics, epidemics, computational algorithms, computer and communications networks, inventory systems financial and risk management, are discussed.

* Pre-requisites: *MATH 451 or equivalent knowledge of real analysis. Knowledge of probability at the level of BIOSTAT 601 or MATH 525.

### Statistics 621: Theory of Probability II

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.

* Pre-requisites: *STATS 520 or equivalent course in measure theory, STATS 620.

### Statistics 625: Probability and Random Processes I (MATH 625)

Axiomatics; measures and integration in abstract spaces. Fourier analysis, characteristic functions. Conditional expectation, Kolmogoroff extension theorem. Stochastic processes; Wiener-Levy, infinitely divisible, stable. Limit theorems, law of the iterated logarithm.

* Pre-requisites: *MATH 597. Graduate standing.

### Statistics 626: Probability and Random Processes II (MATH 626)

Selected topics from among: diffusion theory and partial differential equations; spectral analysis; stationary processes, and ergodic theory; information theory; martingales and gambling systems; theory of partial sums.

* Pre-requisites: *STATS 625. Graduate standing.

### Statistics 630: Topics in Applied Probability

Advanced topics in applied probability, such as queueing theory, inventory problems, branching processes, stochastic difference and differential equations, etc. The course will study one or two advanced topics in detail.

* Pre-requisites:* Permission of instructor.

### Statistics 631: Advanced Time Series Analysis

* Pre-requisites:* STATS 500, 610, 611. Graduate standing.

### Statistics 640: Multivariate Statistical Models (BIOS 890)

Wishart distribution, multivariate linear models, multivariate regression, Hotelling's T-square and its applications, discriminant analysis, canonical correlations, principal components analysis, growth curves.

* Pre-requisites: *MATH 417 and either STATS 611 or BIOSTAT 602. Graduate standing and permission of instructor.

### Statistics 642: Linear Statistical Models I (BIOS 851)

Gauss-Markov theorem; one-way, two-way analysis of variance, and complete higher-way layouts; regression; the general linear model and hypothesis; least squares theory; analysis of covariance; missing observations; multiple comparisons procedures; incomplete blocks, split plot designs, and Latin squares; variance component models, mixed models; treatment of residuals; robustness of the methods. Special topics in the second semester.

* Pre-requisites: *MATH 417 and either STATS 611 or BIOSTAT 602. Graduate standing.

### Statistics 670: Advanced Design and Analysis of Experiments

This is an advanced course on the design and analysis of experiments. It will cover topics from orthogonal arrays, optimal designs, minimum aberration designs, parameter design, response surface methodology, computer experiments, and experiments with split-plot structure. Emphasis will be placed on new concepts/tools and recent advances.

* Pre-requisites: *STATS 570 or permission of instructor.

### Statistics 680: Theory of Sampling

Recent developments in the foundations and methodology of sampling finite populations. Identifiability of units, likelihood of units, likelihood functions, admissibility of standard estimators, randomization, use of prior information in design and inference. Models for non-sampling errors including bias, response error and non-response. Other topics of current interest.

* Pre-requisites: *STATS 426 and 575. Graduate standing.

### 700-Level Statistics Courses

### Statistics 700: Special Topics in Applied Statistics I

Selected topics in applied statistics.

* Prereq: *STATS 501 and graduate standing.

### Statistics 701: Special Topics in Applied Statistics II

Selected topics in applied statistics.

* Prereq: *STATS 501 and graduate standing.

### Statistics 710: Special Topics in Theoretical Statistics I

Selected topics in theoretical statistics.

* Prereq: *Graduate standing and permission of instructor.

### Statistics 711: Special Topics in Theoretical Statistics II

Selected topics in theoretical statistics.

* Prereq: *Graduate standing and permission of instructor.

### Statistics 725: Topics in Advanced Probability I (MATH 725)

* Prereq: *STATS 626. Graduate standing

### Statistics 726: Topics in Advanced Probability II (MATH 726)

* Prereq:* STATS 626, 725; MATH 725. Graduate standing.

### Statistics 750: Directed Reading

Designed for individual students who have an interest in a specific topic (usually that has stemmed from a previous course). An individual instructor must agree to direct such a reading, and the requirements are specified when approval is granted.

* Prereq: *Graduate standing (INDEPENDENT).

### 800-Level Statistics Courses

### Statistics 808: Seminar in Applied Statistics I

** Prereq: **Graduate standing.

### Statistics 809: Seminar in Applied Statistics II

** Prereq: **Graduate standing.

### Statistics 810: Literature Proseminar I

This course is designed to acquaint students with classical papers in mathematics and applied statistics and probability theory, to encourage them in critical independent reading and to permit them to gain pedagogical experience during the course of their graduate training.

** Prereq: **Graduate standing and permission of instructor.

### Statistics 811: Literature Proseminar II

This course is designed to acquaint students with classical papers in mathematics and applied statistics and probability theory, to encourage them in critical independent reading and to permit them to gain pedagogical experience during the course of their graduate training.

** Prereq: **Graduate standing and permission of instructor.

### Statistics 816: Interdisciplinary Seminar in the Physical Sciences

The seminar will consider statistical questions that arise in the physical sciences. Topics will be drawn from current research projects, will vary each semester. Meetings will feature lectures by faculty from the University and selected visitors. Students will be expected to complete a course project and present to group.

** Prereq: **Graduate standing and permission of instructor.

### Statistics 817: Interdisciplinary Seminar in Quantitative Social Science Methodology (EDUC 817/PSYCH 817/SOC 810)

This seminar will meet to consider methodological issues that arise in research in the social sciences. Themes for each meeting will arise from ongoing research projects at the University of Michigan. Visiting researchers will provide a brief account of their aims and data before defining the methodological challenge for which they desire discussion.

** Prereq: **Graduate level courses in Statistics at the level of STATS 500 and 501 or permission of instructor.

### Statistics 818: Seminar in Theoretical Statistics I

** Prereq: **Graduate standing.

### Statistics 819: Seminar in Theoretical Statistics II

** Prereq: **Graduate standing.

### 900-Level Statistics Courses

### Statistics 990: Dissertation/Precandidate

### Statistics 995: Dissertation/Candidate