1. The scientific method (2 lectures)
The role of statistical analysis in science. Model building, prediction,
scientific induction, decission making.
2. Convergence of random vectors (6)
Review of basic convergence concepts. Skorokhod construction.
Multivariate delta-method.
Kolmogorov-Smirnov theorem. Convergence of sample quantiles.
3. Comparison of estimators (3)
Asymptotic relative efficiency.
Sufficiency.
Lower bounds on the variance of estimators. First order efficiency.
3. Methods of maximum likelihood (14)
Consistency of maximum likelihood estimators. Asymptotic normality of
likelihood equation estimators.
The method of scoring. The EM- and IP-algorithms and their properties.
Nonparametric maximum likelihood.
Conditional and partial likelihood.
Monotone likelihood ratio and UMP tests. LMP tests.
The asymptotic distribution of
likelihood ratio, score, and Wald tests (including power at contiguous
alternatives).
4. Improved asymptotic distributions (4)
Edgeworth expansions. Saddlepoint approximations.
5. The likelihood principle (2)
Birnbaum's theorem. Variants and consequences of the likelihood principle.
6. Bayes methods (8)
Conjugate priors. Jeffreys priors. Consistency of Bayes estimates.
Asymptotic normality of posterior distributions. Computation and
approximation strategies. Empirical Bayes methods.
7. Statistical functionals (4)
Differentiation of functionals. Analysis of remainder terms. Asymptotic
properties.
8. Finite sample estimates of variability (5)
Jackknife. Bootstrap. Other resampling methods.
9. Robustness (9)
Qualitative robustness and
resistance. M-estimates. Influence curve and breakdown point.
Minimum distance estimates.
Robustness and Bayes methods.
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10. Estimation theory for dependent data (6)
Likelihood theory for discrete time processes. Consistency and
asymptotic normality. Likelihood tests for Markov chains.
Robustness against dependence of iid-based estimators.
The following topics are possible options. Different instructors will have different points of view. About three of these topics can be covered.
12. Optimal estimation (6)
Minimum variance estimation. Rao-Blackwell theorem. Lehmann-Scheffe theorem.
U-estimates.
Invariance and equivariance. Parameters of location and scale. Pitman
estimates.
13. Nonparametric methods (10)
Invariance. Rank tests based on scores. R-estimates.
L-estimates. Goodness of fit tests. Density estimation. Graphical
methods.
14. Bayesian model comparison (4)
Bayes factors. Inference in the presence of many competing non-nested models.
Comparison between Bayes factors and P-values.
15. Ancillarity in exponential families (6)
Exponential family theory. B-ancillarity, M-ancillarity. Plausibility
inference. Conditional inference.
16. Testing theory (8)
Unbiased and similar tests. Invariance, maximal invariants, UMPI tests.
Local tests in the presence of nuisance parameters. Separate families.
17. Decision theory (8)
Admissibility. Optimality of Bayes procedures. Minimax theory. Shrinkage
estimators.
There is no text available that covers the material we are interested in. The following books each cover some aspect of the course.