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Suggested Texts

• Corcoran, The Simple and Infinite Joy of Mathematical Statistics, 1st edition

• Casella and Berger, Statistical Inference, 2nd edition

• Ross, A First Course in Probability, 9th edition

• Hogg, McKean and Craig, Introduction to Mathematical Statistics, 8th edition

Syllabus

Probability Theory core material:

• Probability:

  • Probability axioms, independence, counting, permutations and combinations

  • Random variables, cumulative distribution functions, probability mass functions, probability density functions, joint distributions, expectation, variance

  • Bernoulli, binomial, geometric, Poisson, uniform, normal, gamma, beta and exponential distributions, multivariate normal

  • Conditional probability, conditional distributions, conditional expectation, conditional variance

• Limit theorems:

  • Modes of convergence (distribution, probability, almost sure, pth mean)

  • Weak and strong law of large numbers

  • Central limit theorems

  • Slutsky鈥檚 theorem, delta method

Mathematical Statistics core material:

• Basics

  • Taylor expansion and multivariate Taylor expansions

  • Transformations of random variables

  • Multivariate transformations

  • Order statistics, minima and maxima

  • Moment generating functions, characteristic functions

  • Exponential families

• Estimation

  • Bias, mean squared error, absolute error

  • Method of moments

  • Maximum likelihood, asymptotic properties, invariance

  • Cramer-Rao lower bound

  • Asymptotic efficiency

  • Uniformly minimum variance unbiased estimators

  • Sufficiency, completeness, Basu鈥檚 theorem, Pitman-Koopman lemma

  • Rao-Blackwell theorem

  • Lehmann-Scheffe theorem

  • Confidence intervals

  • Hypothesis testing, size, power, p-values

  • Uniformly most powerful tests

  • Likelihood ratio tests

  • M-estimators, robust methods

  • EM algorithm applied to mixture models

  • Bayesian statistics: priors, posteriors

  • Applications to linear regression