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Contents
Contents
School of Mathematical and Computer Sciences
STAT354
Distribution Theory
part 1
Statistical Inference
part 2
Printed at the University of New England, October 31, 2000
Contents
Details of the unit
Coordinator
Objectives
Content
Textbook
Timetable
Assessment
Acknowledgements
Distribution Theory
Prerequisites
Probability concepts assumed known
Assumed knowledge of matrices and vector spaces
Preliminaries
Introduction
Indicator Functions
Distribution Functions (cdf's)
Bivariate and Conditional Distributions
Conditional Mean and Variance
Stochastic Independence
Moment Generating Functions (mgf)
Multinomial Distribution
Transformations
Introduction
Bivariate Transformations
Multivariate Transformations (One-to-One)
Multivariate Transformations Not One-to-One
Convolutions
General Linear Transformation
Multivariate Normal Distribution
Bivariate Normal
Multivariate Normal (MVN) Distribution
Moment Generating Function
Independence of Quadratic Forms
Distribution of Quadratic Forms
The role of c.g.f.
Cochran's Theorem
Order Statistics
Introduction
Distribution of Order Statistics
Marginal Density Functions
Joint Distribution of
and
The Transformation
Examples
Non-central Distributions
Introduction
Distribution Theory of the Non-Central Chi-Square
Non-Central t and F-distributions
POWER: an example of use of non-central F
S-Plus commands
Statistical Inference
Reduction of Data
Types of inference
Frequentist inference
Sufficient Statistics
Factorization Criterion
The Exponential Family of Distributions
Likelihood
Information in a Sample
Estimation
Some Properties of Estimators
Unbiasedness
Mean Square Error
Consistency
Efficiency
Cramér-Rao Lower Bound
Minimum Variance Estimation
When can the MVB be Attained?
Properties of Maximum Likelihood Estimates
Interval Estimates
Pivotal Method
Confidence Interval for Population Quantile
Hypothesis Testing
Basic Concepts and Notation
Introduction
Power Function and Significance Level
Relation between Hypothesis Testing and Confidence Intervals
Randomized Tests
Evaluation of and Construction of Tests
Unbiased and Consistent Tests
Certain Best Tests
Neyman Pearson Theorem
Uniformly Most Powerful (UMP) Test
Likelihood Ratio Tests
Background
The Likelihood Ratio Test Procedure
Some Examples
Asymptotic Distribution of
ASSIGNMENTS
Assignments
Assignment 1
Assignment 2
Assignment 3
Assignment 5
Assignment 6.
Past Exams
1997 exam
1998 exam
1999 Exam
About this document ...
Bob Murison 2000-10-31
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