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

$$J({\bf x};H_0,T) = {p_x({\bf x}\vert H_0) \over p_z({\bf z}\vert H_0)}.$$ (2.25)

The numerator density is usually of a simple form so it is known exactly. The denominator density $p_z({\bf z}\vert H_0)$ 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].