27. Totally positive families. A family of distributions with probability densities Pe(x), 8 and x real-valued and varying over n and !!£ respectively, is said to be totally positive of order r (TPr ) if for all XI < . . . < x; and 81 < . . . < 8" (33) s, =IPe,(xd Pe.(xI) Pe,(x,,) Pe (x ) I ° . " for all n=I,2, ... ,r.

It is said to be strictly totally positive of order r (STP,) if strict inequality holds in (33). The family is said to be (strictly) totally positive of order infinity if (33) holds for all n = 1,2, .. .. These definitions apply not only to probability densities but to any real-valued functions P8(x) of two real variables. (i) For r = 1, (33) states that Pu(x) 0; for r = 2, that Pu(x) has monotone likelihood ratio in x. (ii) If a(8) > 0, b(x) > 0, and Pu(x) is STP" then so is a(8)b(x)pu(x). (iii) If a and b are real-valued functions mapping 0 and !l' onto 0' and !l" and are strictly monotone in the same direction, and if Pe (x) is (S)TP" then P8.(x') with 8' =

a- I(8) and x' = b-I(x) is (S)TP, over (O' ,!l").