29. STP3 • Let 8 and x be real-valued, and suppose that the probability densities Pe ( x) are such that Pe :( x )jPe ( x) is strictly increasing in x for 8 < 8' . Then the following two conditions are equivalent:

(a) For 8. < 82 < 83 and k l , k 2 , k3 > 0, let g(x) = klpu,(x) - k 2PU2 ( X) + k 3PU3 ( X). If g( x.) = g( X3) = 0, then the function g is positive outside the interval (XI' X3) and negative inside.

(b) The determinant 6.3 given by (33) is positive for all 8. < 82 < 83 , Xl < 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 XI < X2 < X3 by considering the determinant g(XI) PB2( XI) PB3( XI) g(X2) PB2( x 2 ) PBJX2) g(X3) PB2( X3) PB3( X3) Suppose conversely that

(a) holds. Monotonicity of the likelihood ratios implies that the rank of A3 is at least two, so that there exist constants k1, k 2, k 3 such that g(xI) = g(x3) = O. That the k's are positive follows again from the monotonicity of the likelihood ratios.