I am looking for the name and some reference of the following particular distribution family. Suppose the CDF is $F(x)$, then it has the following property:$$1-F(x;\theta)=[1-F(x)]^\theta.$$
Intuitively, F(x) is something like $1-(1-H(x))^\theta$. Someone suggested that it is called proportional hazard distribution. But I hardly saw any reference on that. Is there any particular name of the family of probability distributions that satisfy this property?
I think the exponential distribution will satisfy the requirement.
$F(x) = 1-e^{-x}$.
$1-F(x;\theta) = 1-(1-e^{-\theta x}) = e^{-\theta x}$.
$[1-F(x)]^\theta = [1-(1-e^{-x})]^\theta = (e^{-x})^\theta = e^{-\theta x}$