Statistical Intervals, 2nd Edition

Appendix G Distribution-Free Intervals Based on Order Statistics

Introduction

This appendix provides technical explanations for the distribution-free statistical intervals based on order statistics, which are described in Chapter 5. The purpose is to provide information such that an interested reader could understand the theoretical basis for the methodology.

The topics discussed in this appendix are:

• Some basic results used to develop and evaluate the coverage probabilities of distribution-free statistical intervals (Section G.1).
• Justification of the distribution-free confidence intervals (and corresponding one-sided confidence bounds) for a distribution quantile (Section G.2).
• Justification of the distribution-free tolerance intervals to contain a given proportion of a distribution (Section G.3).
• Justification of the distribution-free statistical intervals to contain a specified ordered observation in a future sample (Section G.4).
• Justification of the distribution-free statistical intervals to contain at least a given number of observations from a future sample and the relationship of this prediction problem to the prediction of a future ordered observation (Section G.5).

G.1 Basic Statistical Results Used in This Appendix

In this appendix we use the following results:

1. If X is a continuous random variable with and one defines then W ∼ UNIF(0, 1) (see Example D.2 for ...