I want to compute the general formula for the difference between two random variables that follow an Exponentiated distribution.
It is well known that the difference between two independent random variables, that follow a Gumbel distribution, follows a Logistic distribution.
It is also well known that the difference between two independent random variables, that follow an Exponential distribution, follow a Laplace distribution.
More recently, for example, Srivastave, Nadarajah and Kotz (2006) shows that the difference between two independent random variables, that follow an Exponentiated Exponential distribution, follow a Generalized Laplace distribution.
Gumbel and Exponentiated Exponential are particular cases of Exponentiated Distributions: $$ F(x,\theta)=G(x)^{\theta} $$ $$ f(x,\theta)=\theta g(x)G(x)^{\theta-1} $$ Exponential is a particular case of reversed Exponentiated Distribution: $1-F(x,\theta)=[1-G(x)]^{\theta}$
The particular form of such distributions suggests that there should be a more general formula for the difference between two such variables, but I cannot find it or solve it myself. If I use the convolution formula, for $z=x-y$, I get, $$ f_z(z)= \int f_x(x)f_y(x-z) dx = \int \theta g(x) G(x)^{\theta -1} \theta g(x-z) G(x-z)^{\theta -1} dx $$ Maybe with some change of variables this integral can be solved for a general CDF $G(x)$?