Extension of Lemma 3.4.2. Let P0 and P1 be two distributions with densities p0, p1 such that p1(x)/p0(x) is a nondecreasing function of a real-valued statistic T (x).

(i) If T = T (X) has probability density p i when the original distribution of X is Pi , then p 1(t)/p 0(t) is nondecreasing in t.

(ii) E0ψ(T ) ≤ E1ψ(T ) for any nondecreasing function ψ.

(iii) If p1(x)/p0(x) is a strictly increasing function of t = T (x), so is p 1(t)/p 0(t), and E0ψ(T ) < E1ψ(T ) unlessψ[T (x)]is constant a.e.(P0 + P1) or E0ψ(T ) =

E1ψ(T ) =±∞.

(iv) For any distinct distributions with densities p0, p1, −∞ ≤ E0 log  p1(X)
p0(X)

< E1 log  p1(X)
p0(X)

≤ ∞.
[(i): Without loss of generality suppose that p1(x)/p0(x) = T (x). Then for any integrable φ, φ(t)p 1(t) dv(t) = φ[T (x)]T (x)p0(x) dμ(x) = φ(t)t p 0(t) dv(t), and hence p 1(t)/p 0(t) = t a.e.
(iv): The possibility E0 log[p1(X)/p0(X)]=∞ is excluded, since by the convexity of the function log, E0 log  p1(X)
p0(X)

< log E0  p1(X)
p0(X)

= 0.
Similarly for E1. The strict inequality now follows from (iii) with T (x) = p1(x)/p0(x).]