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:
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 Zs = Z(S*; s, ν) is a GPQ for a scalar parameter θ = θ(ν) if it satisfies the following two conditions:
- For given s, the distribution of Zs does not depend on unknown parameters.
- Evaluating Zs at S* = s gives Zs = Z(s; s, ν) = θ.
For the purposes of implementation, we can consider s as the estimate of the parameters ν and that the objective ...