How can I prove (or disprove) that the following function is convex on $X$. $$\rho(X,Z) = Max \{ F^{-1}_Z(t)-F^{-1}_X(t),0 \},$$ where $F^{-1}_X(t)= inf \{ x : F_X(x) \geq t \}$ with $0 \le t \le 1$. Here $F_X(x)=P(X \le x)$, is the CDF of $X$.
If it is convex then we need to prove: $\rho( \lambda X + (1-\lambda)Y,Z) \le \lambda\rho(X,Z) + (1-\lambda) \rho(Y,Z) $.
I dont know how to proceed. It will be helpful if i can get some guidance.