Conditional distributions and independence. Multivariate transformations. Covariance and correlation.
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Mixture distributions. Order statistics. Sums of random variables from a random sample.
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The sampling distributions: Student's t and Snedecor's F. Sufficiency, minimal sufficiency, ancillarity, completeness, Basu's Theorem, Rao-Blackwell Theorem, Lehmann-Scheff Theorem, minimum variance unbiased estimation. Method of moments, maximum likelihood estimation, Bayes' estimation, invariant estimation, consistency. Asymptotic properties of maximum likelihood estimators, the "delta method" for functions of random variables. Hypothesis testing: concepts of significance and power, the Neyman-Pearson lemma, likelihood ratio tests, Bayesian tests.
Interval estimation: methods of finding confidence intervals including inverting a test statistic and pivotal quantities. Bayesian credibility intervals. Statistical Inference.
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Wadsworth [Chapters ]. Linear models: Least squares estimators and their properties. Simple linear regression, multiple regression, Gauss-Markov theorem. Analysis of variance. Linear models with general covariance. Distribution of estimators. General linear hypothesis: F-test and t-test, prediction and confidence regions.
Categorical and continuous covariates, collinearity, interactions. Multivariate normal and chi-squared distributions. Model selection, residual analysis, detecting influential observations, testing for lack of fit, transformations, weighted least squares, and variable selection techniques. Applied Linear Regression. New York: Wiley. Generalized linear models: Exponential families.
Link function, variance function. Iteratively reweighted least squares, asymptotic distribution of maximum likelihood estimators. Analysis of deviance, goodness of fit tests. Log-linear models, analysis of contingency tables, overdispersion.
Logistic regression, case-control studies, multinomial regression. Gamma models. Quasi-likelihood models. Mixed models. Bayesian estimation.
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Generalized Linear Models. Contact Us Seminars. Available Positions Postdoctoral fellowship s. Exams, Applied Math Comp. Exams, Pure Mathematics Comp. Exams, Statistics Comp. Preliminary Examinations The Department of Mathematics and Statistics requires all doctoral students to pass two preliminary examinations, first the Part A and then the Part B. Old Part A Examinations Old examinations are available in the library; they go back until MATH — Ph. Schedule of Examination Each graduate student is allowed only two official attempts at the Part A examination. Exceptions Normally, a student who fails at the first attempt will be required to re-take and pass the entire examination, although in exceptional cases the committee may allow the student to take only one part.
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Applied Mathematics Version The compulsory part of the Paper alpha examination requires students to answer 3 out of 4 questions on Single Variable Real Analysis and 3 out of 4 questions on Linear Algebra. Pure Mathematics Version The compulsory part of the Paper alpha examination requires students to answer 3 out of 4 questions on Single Variable Real Analysis and 3 out of 4 questions on Linear Algebra. Statistics Version The Statistics Part A exam consists of a Theory Paper in mathematical statistics and measure theoretic probability and a Methodology Paper in linear models and generalized linear models.
Syllabus for Paper Alpha — Applied and Pure The compulsory part of this examination requires students to answer 3 out of 4 questions on Single Variable Real Analysis and 3 out of 4 questions on Linear Algebra. Reference: S. Marsden and A.
Tromba, Vector Calculus, W. Freeman, 4th edition Boyce and R. Marsden, Basic Complex Analysis, W. Freeman and Co, 2nd edition Bak and D. Newman, Complex Analysis, Springer Verlag Syllabi for Paper beta modules — Applied Mathematics Discrete Mathematics: Graph theory: trees and cycles; matching theory; connectivity; planar graphs: Kuratowski's theorem, crossing number; graph colouring; perfect graphs; regularity lemma; graph minors; tree-width; random graphs and the probabilistic method. References: Graph Theory , second edition, R.
Diestel, Springer-Verlag, A Course in Combinatorics , second edition, J. Wilson, Cambridge, Trefethen and D. Lectures , , , , Quateroni, R. Sacco, F. Numerical Mathematics, Springer, Sections 7. Hairer, S. Norsett and G. Sections II. III 1,2,3,4.
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Hairer and G. Sections IV. Chapters 7, 8, 13, Optimization: Continuous Optimization Math : Line search methods for unconstrained optimization including step size analysis. Nocedal, S. Wright, Springer, Linear and Nonlinear Programming, D.
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