Suggested Texts
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Corcoran, The Simple and Infinite Joy of Mathematical Statistics, 1st edition
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Casella and Berger, Statistical Inference, 2nd edition
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Ross, A First Course in Probability, 9th edition
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Hogg, McKean and Craig, Introduction to Mathematical Statistics, 8th edition
Syllabus
Probability Theory core material:
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Probability:
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Probability axioms, independence, counting, permutations and combinations
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Random variables, cumulative distribution functions, probability mass functions, probability density functions, joint distributions, expectation, variance
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Bernoulli, binomial, geometric, Poisson, uniform, normal, gamma, beta and exponential distributions, multivariate normal
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Conditional probability, conditional distributions, conditional expectation, conditional variance
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Limit theorems:
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Modes of convergence (distribution, probability, almost sure, pth mean)
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Weak and strong law of large numbers
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Central limit theorems
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Slutsky鈥檚 theorem, delta method
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Mathematical Statistics core material:
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Basics
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Taylor expansion and multivariate Taylor expansions
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Transformations of random variables
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Multivariate transformations
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Order statistics, minima and maxima
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Moment generating functions, characteristic functions
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Exponential families
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Estimation
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Bias, mean squared error, absolute error
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Method of moments
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Maximum likelihood, asymptotic properties, invariance
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Cramer-Rao lower bound
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Asymptotic efficiency
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Uniformly minimum variance unbiased estimators
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Sufficiency, completeness, Basu鈥檚 theorem, Pitman-Koopman lemma
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Rao-Blackwell theorem
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Lehmann-Scheffe theorem
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Confidence intervals
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Hypothesis testing, size, power, p-values
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Uniformly most powerful tests
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Likelihood ratio tests
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M-estimators, robust methods
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EM algorithm applied to mixture models
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Bayesian statistics: priors, posteriors
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Applications to linear regression
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