Projection Method for Moment Bounds on Order Statistics from Restricted Families: I. Dependent Case

We present a method of projections onto convex cones for establishing the sharp bounds in terms of the first two moments for the expectations ofL-estimates based on samples from restricted families. In this part, we consider the case of possibly dependent identically distributed parent random variables. For the classes of decreasing failure probability, DFR, and symmetric unimodal marginal distributions, we first determine parametric subclasses which contain the distributions attaining the extreme expectations for allL-estimates. Then we derive the bounds for single order statistics. The results provide some new characterizations of uniform and exponential distributions