Maths 321: Mathematical Statistics - II (4)

Syllabus

1.  Prerequisite:  Maths 320. 

2.  Course Description: Point and interval estimation, maximum likelihood, Neyman-Pearson Lemma, Likelihood ratio tests, classical tests of significance, goodness-of-fit, contingency tables, correlation, regression, nonparametric methods, Bayesian methods.

3.  Course Objectives: The theory of statistics can be treated as a branch of mathematics in which probability is the basic tool.  This course familiarizes students with the tree steps a statistician faces with a real-life problem in statistics.  First, how to select a mathematical model.  Second, how to check the reasonableness of the model.  Third, how to draw proper conclusions from the model to solve the proposed problem.  The course considers both theory and applications, although the emphasis is on the theory.

4.  Course Rationale: Mathematical statistics is the study of how to deal with data by means of probability models.  It grew out of methods for treating data that were obtained by some repetitive operation such as those encountered in games of chance and in industrial processes.  These methods soon found application in such diverse fields as medical research, insurance, marketing, agriculture, chemistry, and in indurstial experimentation.  This course is, therefore, a required course for statistics and actuarial science majors and is an excellent course for those who may need statistical methods in their work at a reasonably sophisticated level.

5.  Course Content:

            Estimation

                        Maximum Likelihood Estimation

                        Properties of Estimators

                        Confidence Intervals for Means

                        Confidence Intervals for Variances

                        Confidence Intervals for Proportions

                        Sample Size

                        Sufficient Statistics

                        Chebyshev’s Inequality

            Tests of Statistical Hypotheses

                        Critical Region

                        Type I and type II Errors

                        Power

                        Best Critical Regions

                        Neyman-Pearson Theorem

                        Likelihood Ratio Tests

                        Tests About Means and Proportions

                        Test About Variance

                        Tests About Difference of Means

            Nonparametric Methods

                        Order Statistics

                        Confidence Intervals for Percentiles

                        Binomial Tests for Percentiles

                        Wilcoxon Test

                        Two-Sample Distribution-Free Tests

                        Run Tests and Tests for Randomness

                        Kolmogorov-Smirnov Goodness-of-Fit Test

            Chi-square Tests for Models

                        Basis Chi-square Statistic

                        Testing Probabilistic Models

                        Comparisons of Several Distributions

                        Contingency Tables

            Linear Statistical Models

                        Simple Regression

                        Tests of Equality of Means

                        Two-Factor Analysis of Variance

            Bayesian Decision Theory

                        Compound distributions

                        Decision Theory

                        Bayesian Methods

6.  Course Format: Lecture/discussion.  The amount of material to be covered may not allow for a complete treatment in class of all topics listed in the Course Content, so students may need to supplement overviews in class with individual reading.      

7.  Methods of Evaluating Student Performance: Course grades are determined primarily on student performance of tests and the final examination, augmented by evaluation of performance on homework.  

8.  Evaluation of the Course: The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Graduate Programs Committee.  

Ali /rev. Nelson 2002

M. Begum 10/17/05