Statistics based on empirical distribution

Question

Let $\beta:[0,1]\rightarrow[0,\infty)$ and $\alpha > 0$ and $X_{1},...,X_{n}$ a random sample from a distribution with continuous c.d.f $F$ and $\hat{F}_n$ the empirical distribution function. We define statistics: $V_{\beta, \alpha} = \int \beta(\hat{F}_n(x))|\hat{F}_n(x)-F(x)|^{\alpha}\hat{F}_n(dx)$. How to show that the distribution of $V_{\beta, \alpha}$ does not depend on $F$?