  A location- and scale-invariant predictor is constructed which exhibits good
probability matching for extreme predictions outside the span of data drawn
from a variety of (stationary) general distributions. It is constructed via the
three-parameter {\mu, \sigma, \xi} Generalized Pareto Distribution (GPD). The
predictor is designed to provide matching probability exactly for the GPD in
both the extreme heavy-tailed limit and the extreme bounded-tail limit, whilst
giving a good approximation to probability matching at all intermediate values
of the tail parameter \xi. The predictor is valid even for small sample sizes
N, even as small as N = 3.
  The main purpose of this paper is to present the somewhat lengthy derivations
which draw heavily on the theory of hypergeometric functions, particularly the
Lauricella functions. Whilst the construction is inspired by the Bayesian
approach to the prediction problem, it considers the case of vague prior
information about both parameters and model, and all derivations are undertaken
using sampling theory.
