Estimates for order statistics in terms of quantiles

Let $X_1,\dots, X_n$ be independent non-negative random variables with cumulative distribution functions $F_1,F_2,\dots,F_n$, each satisfying certain (rather mild) conditions. We show that the median of $k$-th smallest order statistic of the vector $(X_1,\dots,X_n)$ is equivalent to the quantile of order $(k-1/2)/n$ with respect to the averaged distribution $F=\frac1n\sum_{i=1}^n F_i$.