Statistics Preliminary Exam Syllabus

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