Chapter 2 STAT 205B: Intermediate Classical Inference

This chapter contains past exam problems of STAT 205B: Intermediate Classical Inference. The textbooks used for this class is Casella and Berger (2002). A possible syllabus is listed here.

• Sufficient and minimal sufficient statistics. Finding a sufficient statistic and showing that it is minimal sufficient. The factorization theorem. Examples from common distributions. Sufficiency and the exponential family. Ancillary statistics. Complete statistics. Techniques to find complete sufficient statistics. Examples mainly from exponential family. Basu’s theorem connecting sufficiency, completeness and ancillary properties.

• Principles of statistical inference: Sufficiency, conditionality and likelihood.

• Methods of finding estimators: method of moments; maximum likelihood estimation; unbiased estimators. Definitions and properties; examples with common distributions. Possible logical inconsistency of frequentist estimators (e.g., a negative estimate of a positive parameter).

• Methods for evaluating estimators: mean squared error, minimum variance, the Rao-Blackwell theorem, the Lehman-Scheffe theorem, the Cramer-Rao inequality.

• Methods for finding tests: likelihood-ratio tests.

• Methods for evaluating tests: error probabilities. Neyman-Pearson lemma for point null vs. point alternative. Extend it to composite null vs. composite alternative by using monotone likelihood ratio (MLR) property. UMP tests, p-values.

• Methods for finding interval estimators: Pivotal quantities, inverting a test statistic.

• Methods for evaluating interval estimators: coverage probability and size, illustrate in location and scale families how to find intervals with shortest length.

• Large sample theory for estimation, testing and confidence intervals.

References

Casella, George, and Roger Berger. 2002. Statistical Inference. 2nd ed. Belmont, CA: Duxbury Resource Center.