Non-positive upper bounds on expectations of low rank order statistics from DFR populations

Abstract

Danielak (Sharp upper mean-variance bounds on trimmed means from restricled families, Statistics 37 (2003), pp. 305–324) established strictly positive optimal upper mean-variance bounds on the expectations of order statistics with relatively large ranks coming from decreasing failure rate populations. She also proved that the respective evaluations of low rank order statistics cannot be positive. Here, we show that the zero bounds on the deviation of the expected small order statistic from the population mean expressed in the scale units based on the pth absolute central moments with p>1 cannot be improved, and determine strictly negative sharp bounds in terms of the mean absolute deviation units.