Statistical Intervals, 2nd Edition

Appendix D General Results from Statistical Theory and Some Methods Used to Construct Statistical Intervals

Introduction

This appendix provides some useful tools and results from statistical theory. These tools facilitate the justification and extension of much of the methodology in the book.

The topics discussed in this appendix are:

• Basic theory for transformation of random variables (Section D.1).
• The delta method to obtain expressions for approximate variances of random quantities as a function of the variances and covariances of the function arguments (Section D.2).
• Expected and observed information matrices (Section D.3).
• Some general regularity conditions assumed in most of the book and needed for certain technical results (Section D.3.2).
• A definition of convergence in distribution of random variables, with examples of its use in this book (Section D.4).
• An outline of general maximum likelihood theory relevant to applications in this book (Section D.5).
• The cdf pivotal method for constructing confidence intervals, their coverage probabilities, and examples for continuous and discrete distributions (Section D.6).
• Bonferroni approximate statistical intervals with application to simultaneous confidence intervals as well as the construction of tolerance and simultaneous prediction intervals (Section D.7).

D.1 The cdfs and pdfs of Functions of Random Variables

This section reviews the procedure for obtaining the cdf and pdf (or pmf) of a one-to-one function of a random ...