Contents

• 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 $Y_r$ and $Y_s$
    • The Transformation $F(X)$
    • 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 $-2\log \Lambda$
• ASSIGNMENTS
  • Assignments
  • Assignment 1
  • Assignment 2
  • Assignment 3
  • Assignment 5
  • Assignment 6.
• Past Exams
  • 1997 exam
  • 1998 exam
  • 1999 Exam
• About this document ...