Testing Statistical Hypotheses

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Typical Scheduling: 
Every fall semester

Basic theories of testing statistical hypotheses, including a thorough treatment of testing in exponential class families. A careful mathematical treatment of the primary techniques of hypothesis testing utilized by statisticians.


MATH 4261, MATH 4262 or equivalent, MATH 6241 and MATH 6262

Course Text: 

At the level of Lehmann & Romano, Testing Statistical Hypotheses, 3rd edition, Springer-Verlag

Topic Outline: 
  • Decision Theoretic Context, Geometry of the Risk Set, and Relationship to Classical Testing Criteria
  • Univariate Testing:
    • Neyman-Pearson lemma for simple versus simple hypotheses
    • application to UMP tests for monotone likelihood ratio families
    • exponential families of distributions
    • generalized Neyman-Pearson lemma and application to UMPU single parameter tests
    • locally best tests with applications to rank tests
  • Testing in Multivariate Koopman Darmois Families
    • similarity and completeness
    • application to UMPU tests, including the one and two sample t-tests
    • group invariance of statistical decision problems and its use in testing hypotheses
    • UMPI tests including the general linear model
  • Large Sample Properties: Pitman Efficiency, Applications to Distribution-Free Tests
  • Sequential Testing: Optimality of the SPRT