Sharp lower bounds on expectations of gOS based on DGFR distributions

We present the lower bounds expressed in standard deviation units on the expectations of the generalized order statistics (gOS) which are based on the parent distributions with the decreasing generalized failure rate. The particular cases are families of distributions with the decreasing density and decreasing failure rate. The bounds are obtained with the use of the projection method applied to functions satisfying some particular conditions and appropriately chosen convex cones. We also provide the attainability conditions. The results are illustrated with the numerical results on the progressively type II censored order statistics, which are one of the special cases of gOS.