## Math 334 & Stat 434 at Lehigh University

Math 334/Stat 434 (Mathematical Statistics): Popolations and random sampling; sampling distributions; theory of estimation; criteria and methods of point and interval estimation; theory of testing statistical hypotheses. Prerequisite: Math 231 or Math 309.

Here is a typical course description of Mathematical Statistics at lehigh University.

Text for the Fall of 2001:  Statistics: Theory & Methods, 2nd edition, by Berry and Lindgren.

Texts used in previous years:
1. "Mathematical Statistics and Data Analysis", 2nd edition, by John A. Rice.
2. "Probability & Statistical Inference", by Hogg and Tenis
3. "Statistical Inference", by Casella and Burger
4. "Theoretical Statistics", by Cox and Hinkley

Selected topics :

• Review (sample space, probability measure, conditional probability, independence, randon variables, some known discrete and continuous distributions, joint distributions, expected values, variance, covariance).
• Moments and Moment generating function.
• Can moments always uniquely determine a probability distributions?
• Approximate methods (delta-method and the continuous mapping theorem).
• Limit theorems (Convergence in probability, LLN, Central Limit Theorem, and convergence in distribution).
• Distribution derived from normal distribution (Chi-squared, t, F, and other related distributions).
• Fitting the distribution (minimum chi-squared and other methods).
• Unbiased and consistent estimator.
• Method of moments, consistency of moment estimator.
• Fisher Information and Cramer-Rao Inequality.
• Exact vs Asymptotic variance, delta-method revisit.
• Method of maximum Likelihood, large sample thoery for MLE.
• Empirical Fisher information and approximate confidence interval based on MLE.
• Regular (smooth) probability family.
• Sufficient statistics and factorization theorem.
• Sufficiency and Rao-Blackwell Theorem.
• MVUE (Minimum Variance Unbiased Estimator).
• Complete statistics and Lehmann-Scheffe Theorem.
• Exponential probability family.
• Order statistics.
• Hypotheses testing.
• Probabilities of Type-I and Type-II errors; Power function.
• N-P Lemma.
• Likelihood ratio tests.
• Others