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.
Journal of the Iranian Statistical Society