I am interested in a family of conditional distribution with the following Single-Crossing Property (SCP):
For any given parameter a, there is a unique cutoff point c(a) s.t.
(Partial derivative w.r.t. a) d[G(x|a)]/da>0 for any x<c(a); d[G(x|a)]/da<0 for any x>c(a).
In other words, the partial derivative function w.r.t. a: d[G(x|a)]/da crosses y=0 line from above once and only once. So I call it as a SCP.
Can anybody name a family of conditional distribution behaves in such a way? Hopefully, with monotone cutoff as a function of parameter a, i.e. c'(a)>0(or <0).
Thanks a lot!