STAT 52800 Introduction to Mathematical Statistics
Course Description: STAT 52800 Introduction to Mathematical Statistics
Textbook: Probability for Statisticsand Machine Learning: Fundamentals and Advanced Topics, pdf
Notes:
- problems of inference
- Exponential family
- statistical models: distributions of Exponential family, heavy tails and kewness
- decision theory: evaluating an estimator and risk function, maximum risk and minimaxity
- Exponential families, moments and moment generating function
- point estimations
- likelihood function, MLE, score function
- Fisher information, sufficient statistics and factorization theorem,
- minimal sufficient statistics
- ancillary statistics and MLE in curved Exponential families, Basu’s theorem
- Rao-Blackwell theorem, uniformly minimum variance unbiased estimate (UMVUE)
- completeness, Lehmann-Scheffe theorem,
- asymptotic unbiasedness
- Cramer-Rao inequality
- asymptotic approximations
- consistency of estimator sequence
- Cramer-Rao Conditions for consistency of MLE
- asymptotic distribution of MLE, Fisher information matrix
- pivot, Wald confidence intervals
- Delta theorem of Cramer, variance stabilizing transformations (VST)
- Rao’s score intervals, Wald confidence intervals and Slutsky’s theorem
- testing of hypotheses and confidence regions
- type I and type II error probabilities
- randomized and nonrandomized tests
- most powerful tests, Neyman-Pearson lemma
References:
Stat 609: Mathematical Statistics I, University of Wisconsin-Madison
Statistics IB: University of Cambridge
- see page 14-16 for Rao-Blackwell theorem
notes for Theoretical Statistics, University of Arizona
- see note for sufficient statistic
An Introduction to Probability Theory and Its Applications, Volume 1, William Feller, pdf
An Introduction to Probability Theory and Its Applications, Volume 2, William Feller