Does the distribution of a the second order statistic of a distribution dominate (i.e., FOSD) the distribution of the first order statistic? I assume that it does, but it would be very helpful if a proof already existed or if I could prove it.
Here is the general case. Suppose we have a distribution $G$ with $n$ draws, and we are looking at the $k$th order statistic. The value of this order statistic is distributed $G_{(k)}$. Now consider the $(k-y)$th order statistic ($y$ is a positive whole number smaller than $k$). This order statistic is distributed $G_{(k-y)}$.
Does $G_{(k)}$ FOSD (first-order stochastic dominate) $G_{(k-y)}$? I can't quite figure out how to do this, and if this has already been proven somewhere, that would be great to know about.