### Fixed reference hypothesis

For modules using a fixed reference hypothesis, care must be taken in calculation of the J-function because the data is more often than not in the tails of the PDF. For fixed reference hypotheses the J function is

 (2.25)

The numerator density is usually of a simple form so it is known exactly. The denominator density must be known exactly or approximated carefully so that it is accurate even in the far tails of the PDF. The SPA (Section 2.3.2), provides a solution for cases when the exact PDF cannot be derived, but the exact moment-generating function (MGF) is known. The SPA is known to be accurate in the far tails of the PDF [16].

Baggenstoss 2017-05-19