Abstract
Rychlik [Appl Math (Warsaw) 29:15–32, 2002] presented positive sharp upper bounds on the expectations of order statistics with sufficiently large ranks, based on i.i.d. samples from the decreasing density and failure rate populations (DDA and DFRA, for short). They were expressed in terms of the population mean and standard deviation. Here we provide respective non-positive upper tight evaluations for expected small order statistics centered about the population mean, measured in various scale units.
Highlights
We assume that X1, . . . , Xn are i.i.d. random variables, and X1:n ≤ · · · ≤ Xn:n stand for the respective order statistics
We extend the definition to the distributions with atoms described above in order to make the family of decreasing density on the average distributions (DDA) closed
Positive sharp mean-standard deviation estimates of the expected order statistics and spacings from the DD and DFR families were determined by Danielak (2003) and Danielak and Rychlik (2004), respectively
Summary
We assume that X1, . . . , Xn are i.i.d. random variables, and X1:n ≤ · · · ≤ Xn:n stand for the respective order statistics. The first optimal evaluations of the expectations of order statistics from the decreasing density and failure rate populations (DD and DFR, respectively) are due to Gajek and Rychlik (1998). They were represented in the scale units connected with the second raw moments of the parent distribution. Positive sharp mean-standard deviation estimates of the expected order statistics and spacings from the DD and DFR families were determined by Danielak (2003) and Danielak and Rychlik (2004), respectively. Rychlik (2002) determined positive tight upper mean-standard deviation bounds on the expectations of order statistics with sufficiently large ranks, coming from DDA and DFRA populations. Negative sharp upper evaluations for the expectations of generalized order statistics coming from arbitrary populations were studied in Goroncy (2013)
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