STAT 472
MATHEMATICAL STATISTICS

1.
Sampling From Normal Populations
Distributions of \bar{X} , S^2 , S^2_1 / S^2_2 etc. Confidence intervals for /mu , \sigma^2 , \mu_1 - \mu_2 , \sigma^2_1 / \sigma^2_2 .

2.
Estimation Principles
(a)
The Bayesian method. Prior, likelihood and posterior. Examples: normal, exponential, binomial. Loss functions, Bayes risk, Bayes estimates.
(b)
Maximum likelihood estimators. Some examples.
(c)
Sufficiency, Factorization criterion. Bayes estimates, m.l.e.'s and sufficiency.
(d)
Asymptotic variance of m.l.e., Fisher information, information inequality.
(e)
Unbiasedness. Definition of UMVUE. Some examples.

3.
Tests of Hypotheses
(a)
Basic definitions including power function.
(b)
Neyman-Pearson Lemma. Uniformly most powerful tests.
(c)
Monotone likelihood ratio and UMP tests.

References

1.
W. Mendenhall. D.D. Wackerly and R.L. Scheaffer. Mathematical Statistics with Applications. Duxbury (4th Edition).
2.
R.V. Hogg and A.T. Craig. Introduction to Mathematical Statistics. New York: Macmillan.
3.
M.H. DeGroot. Probability and Statistics. Reading, MA: Addison-Wesley.