Bounds on expectations of small order statistics from decreasing density populations

For i > (n + 1)/2, Danielak (Statistics 37:305–324, 2003) established an optimal positive upper mean-variance bound on the expectation of ith order statistic based on the i.i.d. sample of size n from the decreasing density population. We show that the best bounds on the expected deviation of the ith order statistics from the population mean, i ≤ (n + 1)/2, expressed in more general scale units generated by pth absolute central moments with p > 1 amount to zero. We also determine the respective strictly negative bounds in the mean absolute deviation units.