Sharp upper bounds for the expected values of non-extreme order statistics from discrete distributions

We consider the problem of determining sharp upper bounds on the expected values of non-extreme order statistics based on i.i.d. random variables taking on N values at most. We show that the bound problem is equivalent to the problem of establishing the best approximation of the projection of the density function of the respective order statistic based on the standard uniform i.i.d. sample onto the family of non-decreasing functions by arbitrary N  -valued functions in the norm of L²(0,1)$L^2(0,1)$ space. We also present an algorithm converging to the local minima of the approximation problems.