*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

**is**

*S**F*(·;

**), where**

*ν***is a vector of unknown parameters. Denote an observation from**

*ν***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**

*S**Z*=

_{s}*Z*(

***;**

*S***,**

*s***) is a GPQ for a scalar parameter**

*ν**θ*=

*θ*(

**) if it satisfies the following two conditions:**

*ν*- For given
, the distribution of*s**Z*does not depend on unknown parameters._{s} - Evaluating
*Z*at_{s}* =*S*gives*s**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

**and that the objective ...**

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