Statistical Intervals, 2nd Edition

Appendix B Generic Definition of Statistical Intervals and Formulas for Computing Coverage Probabilities

B.1 Introduction

This appendix provides formal definitions of two-sided confidence intervals, tolerance intervals, prediction intervals, and the corresponding one-sided bounds. A procedure to compute a statistical interval (or bound) defines a computational method by which interval endpoint(s) are obtained from observed data. This appendix also provides methods to compute the “coverage probability” (CP) associated with a procedure for calculating a statistical interval, the CP being is the probability that the interval obtained using the procedure actually contains what it is claimed to contain, as a function of the procedure’s definition. General expressions are followed by some specific examples that can be programmed directly for purposes of computation. Both analytical and simulation-based methods are described.

Knowledge of the CP of an interval is useful for several purposes:

• In some cases, exact (or approximate) statistical interval procedures can be obtained by finding interval endpoints that result in a CP that is equal to (or approximates) the desired nominal confidence level.
• To evaluate the adequacy of an approximate interval procedure.
• To calibrate a procedure (i.e., to improve the approximation). This topic is discussed in Section B.8 and applied in Chapters 6 and 7.

This appendix describes the definition and CPs for:

• Confidence intervals and one-sided confidence ...