Statistical Estimation

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Every spring semester

Basic theories of statistical estimation, including optimal estimation in finite samples and asymptotically optimal estimation. A careful mathematical treatment of the primary techniques of estimation utilized by statisticians.


MATH 4261, MATH 4262 or equivalent and MATH 6241

Course Text: 

At the level of Lehmann, Theory of Point Estimation

Topic Outline: 
  • Statistical decision theory: geometry of decision problems, the fundamental theorem of game theory and its use in statistical decision theory, specialized techniques for finding minimax and Bayes estimators in standard problems of estimation
  • The Bayesian viewpoint: solving the no-data problem and using it in univariate and multivariate settings, detailed analysis for conjugate priors
  • Optimality under restrictions:
    • Minimum variance unbiased estimation: the Rao-Blackwell and Lehmann-Scheffe theorems
    • Equivariant estimation: invariance of statistical problems under groups and some applications in estimation
  • Asymptotic theory of estimation:
    • General notions of asymptotic optimality: Hodges counterexample
    • Le-Cam's theorem on asymptotic optimality
    • Asymptotic optimality of maximum likelihood estimators, special cases including logistic regression
    • Robust estimators (M, L, and R) and their asymptotic relative efficiencies
    • Asymptotic optimality of Bayes estimators including higher order analysis characterizing asymptotic posterior distributions