Suppose λ,x,ζ traverse the ordered sets Λ, X and Z, respectively and consider the functions f(λ,x,ζ) and g(λ,ζ) satisfying the following conditions,
(a) f(λ,x,ζ) > 0 and f is TP2 in each pairs of variables when the third variable is held ﬁxed and
(b) g(λ,ζ) is TP2.
Then the function
h(λ,x) =∫Z f(λ,x,ζ)g(λ,ζ)dµ(ζ), deﬁned on Λ×X is TP2 in (λ,x). The aim of this note is to use a new stochastic ordering argument to prove the above result and simplify it’s proof given by Karlin (1968). We also prove some other new versions of this result.

Received: 2014/12/16 | Accepted: 2014/12/16 | Published: 2014/12/16