STP3. Let and x be real-valued, and suppose that the probability densities p(x) are such that

STP3. Let θ and x be real-valued, and suppose that the probability densities pθ(x) are such that pθ
(x)/pθ(x) is strictly increasing in x for

θ<θ

. Then the following two conditions are equivalent:

(a) For θ1 < θ2 < θ3 and k1, k2, k3 > 0, let g(x) = k1pθ1 (x) − k2pθ2 (x) + k3pθ3 (x).

If g(x1) − g(x3) = 0, then the function g is positive outside the interval (x1, x3)

and negative inside.

(b) The determinant 3 given by (3.48) is positive for all

θ1 < θ2 < θ3, x1 < x2 < x3. [It follows from

(a) that the equation g(x) = 0 has at most two solutions.]

[That

(b) implies

(a) can be seen for x1,< x2 < x3 by considering the determinant

%

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%

g(x1) g(x2) g(x3)

pθ2 (x1) pθ2 (x2) pθ2 (x3)

pθ3 (x1) pθ3 (x2) pθ3 (x3)

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Suppose conversely that

(a) holds. Monotonicity of the likelihood ratios implies that the rank of 3 is at least two, so that there exist constants k1, k2, k3 such that g(x1) = g(x3) = 0. That the k

s are positive follows again from the monotonicity of the likelihood ratios.]