Instructor:  Paul J. Smith, Statistics Program

Office hours:  MW 3-5 in MTH 1107
Telephone:  (301) 405-5104 or (301) 405-5061
E-mail:  pjs@math.umd.edu

Textbook:   Casella, G. and Berger, R. L. (2002).    Statistical Inference (2nd ed.).   Pacific Grove, CA: Duxbury/Thompson Learning.

Prerequisite:  STAT 700.

Statistics is often defined as the science of gathering, analyzing and interpreting data. Mathematical statistics is the underlying theory which establishes rigorous statistical methods of inference from data and which makes it possible to assess possible errors in these inferences. The theory requires models for the data generating and data collection processes and also models for the sources of error. This modeling is based on probability theory, the mathematics of (random) errors.

STAT 700-701 is a year long course in mathematical statistics. The course will focus on formulation and analysis of the most important and widely used statistical models, including both measurement and count data. While real world data sets will be employed as examples, the emphasis will be on the underlying mathematical techniques. It is not assumed that students have had a course in statistics, but they must have mastered probability at the level of Ross (2002).

STAT 700 began with a review of the necessary probability background. These probability topics were applied immediately to concepts of statistical data reduction, both for the important example of normally distributed data and for general models for data. Next, the course treated general principles for statistical inference and the specific problem of point estimation.

STAT 701 will begin by presenting the closely related topics of hypothesis testing and confidence sets , and then will turn to a systematic treatment of Bayesian statistics and large sample theory. Linear models, including regression and analysis of variance, will be treated from the mathematical point of view. The student should realize, however, that linear models are fundamental in applied statistics. Next, analysis of categorical data will be discussed. Finally, the course will treat topics in modern computational statistics, such as simulation and resampling methods.

The material in STAT 700-701 forms the syllabus for the Written Examinations in Mathematical Statistics, both at the Master's and Ph.D. levels.

STAT 701 Topics:

• Hypothesis Testing:  The two decision problem--type I and type II errors. Most powerful tests of simple hypotheses and the Neyman-Pearson lemma. Composite hypotheses, monotone likelihood ratio property. Applications to exponential families. Likelihood ratio tests.
• Confidence Sets:  Definitions and duality of confidence sets and hypothesis tests. Pivotal quantities, construction of confidence intervals and regions. Examples for normal, binomial and exponential family data.
• Decision Theory and Bayesian Inference:  Elements of decision theory, criteria of goodness, Bayes procedures, admissibility and minimax rules.
• Large Sample Theory:  Consistency, asymptotically normality and asymptotic efficiency; asymptotics of maximum likelihood estimators; estimates of precision; large sample tests; robust estimators, estimating equations and M-estimators.
• Linear Models, Regression and ANOVA:  Examples: simple linear regression and comparing groups, linear models and least squares, Gauss-Markov Theorem, distribution theory with normal errors, extensions.
• Categorical Data and Generalized Linear Models:  Chi-squared tests of fit; regression models for binary and binomial data, generalized linear models.
• Nonparametric Models and Resampling:  Estimating distributions and functionals; functions of order statistics; the resampling paradigm; bootstrap and jackknife for variance estimation and bias adjustment; simulation; Markov Chain Monte Carlo for Bayesian inference.

• Midterms: There will be two in-class midterms, during the fourth and ninth weeks of the semester.
• Final:  Thursday, May 19, 4-6 p.m. The final will be held in MTH 0305, our regular classroom, but note the change in time..
• Homework:  Frequent problem sets will be assigned.  Students should not work together on homework assignments.  Click here for homework assignments.
• Grades:   Final course grades will be based on a weighted average, tentatively with weights of 30% for the final exam, 20% for each midterm and 30% for homework.
Click here for Exam 2 solutions.

Bickel, P. J. and Doksum, K. A.  (1977).  Mathematical Statistics.  San Francisco: Holden-Day.

Hogg, R. V., McKean, J. W. and Craig, A. T.  (2005).  Introduction to Mathematical Statistics (6th ed.).  Upper Saddle River, NJ: Pearson Prentice Hall.

Kiefer, J. C.  (1987).  Introduction to Statistical Inference.  New York: Springer-Verlag.

Rao, C. R. (1973).  Linear Statistical Inference and Its Application.   New York: J. Wiley.

Rohatgi, V. K. (1976).  An Introduction to Probability Theory and Mathematical Statistics.   New York: J. Wiley.

Ross, S. (2002).  A First Course in Probability (6th ed.).   Upper Saddle River, NJ: Prentice-Hall.

Shao, J. (2003).  Mathematical Statistics (2nd ed.).   New York: Springer-Verlag.