List of STAT 700 Topics for Fall 2022 Final Exam
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Eric Slud                                     12/12/2022

The exam will consist of 4 problems (or maybe 5 with choice of 4), 
from 8am to 10am in MTH 0103.

The ground rules will be the same as for the in-class test. You can 
bring a single notebook page of notes as a memory aid. (You may use 
two sides, but I recommend against it.) You can bring a calculator if you like, 
butI will not ask for a numerically simplified answer in any numerical problem. 
(Some answers might be given in terms of quantiles of standard probability 
distributions.)

Here are the topics that we covered heavily this semester and that are in scope 
for the final exam. The order is roughly chronological.

   I. Statistical definitions: Data Structure, (Parametric) Probability models,
      Parameter Space, Identifiability of Parameters, Consistency of Estimators,
      Empirical Distribution Function and estimators defined from it.
   
   II. Bayesian formulation of statistical problems. Posteriors. Conjugate priors.
       Mixture models as hierarchical models.

   III. Decision theory. Definition of non-randomized and randomized decision   
       rules. Risk, Bayesian Decision Theory, Posterior risk, Prediction.
       Minimax decision rules, admissibility.
       Complete Class -- definition, results:
              connected with MLR and one-sided hypothesis test
              class of (possibly randomized) decision rules that are 
                  functions of a sufficient statistic 

   IV. Sufficiency. Factorization Theorem. Rao-Blackwell Theorem.
       Minimal Sufficient Statistics. Complete Sufficient statistics.
       Basu's Theorem. UMVUEs. Information (Cramer-Rao) inequality.

   V. Exponential Families. Sufficient statistics from samples.
      Natural parameter. Rank. Curved exponential families.
      Results for canonical exponential families (with open natural parameter space): 
         Existence and uniqueness of MLEs
         Minimality of sufficient statistic
         Completeness of sufficient statistic: consequence for UMVUEs.
         Representation of MLE as General Method of Moments Estimator
         Conjugate Priors associated with Exponential Family
         Exponential Families characterized by attaining Cramer-Rao Bound
         MLR Property

    VI. Hypothesis Testing. Neyman Pearson Lemma. 
         UMP Tests in MLR problems (one-sided tests).
         Complete Class result for NP type tests in MLR setting