$f(x)$, $p(x)$, $g(y)$ and $q(y)$ are four functions, where $f(x)$ and $g(y)$ are probability density functions. $C$ is a constant.
Now I have some statistics $x_1,...,x_n$ and $y_1,...,y_n$, where $x_i \sim f(x)$ and $y_i \sim g(x)$. $p(x_i)$ and $q(y_i)$ are known for all $i$.
Is there any statistical hypothesis testing method for the hypothesis $f(x)p(x)+g(y)q(y) \equiv C$?
If not, what about a simplified problem where $p(x) \equiv q(y) \equiv 1$?