Statistical Intervals, 2nd Edition

Appendix F Generalized Pivotal Quantities

Introduction

This appendix provides a general definition of a generalized pivotal quantity (GPQ), a method to obtain GPQs as a function of other GPQs, and a set of conditions that when satisfied ensure exact confidence intervals based on a GPQ.

The topics discussed in the appendix are:

• Definition of a GPQ (Secton F.1).
• A substitution method to obtain GPQs (Section F.2).
• Examples of GPQs for functions of parameters from location-scale distributions (Section F.3).
• Conditions for exact intervals derived from GPQs (Section F.4).

F.1 Definition of a Generalized Pivotal Quantity

Let S be a vector function of the observed data (i.e., a vector of parameter estimators). Suppose that the cdf of S is F(·; ν), where ν is a vector of unknown parameters. Denote an observation from S by s and let S* be an independent copy of S. That is, S and S* are independent and have the same distribution. A scalar function Z_s = Z(S*; s, ν) is a GPQ for a scalar parameter θ = θ(ν) if it satisfies the following two conditions:

1. For given s, the distribution of Z_s does not depend on unknown parameters.
2. Evaluating Z_s at S* = s gives Z_s = Z(s; s, ν) = θ.

Note that Hannig et al. (2006, Definition 2) refer to this as a “fiducial generalized pivotal quantity.”

For the purposes of implementation, we can consider s as the estimate of the parameters ν and that the objective ...