AISM 53, 877-894

© 2001 ISM

## Stability of order statistics under dependence

### Tomasz Rychlik

Institute of Mathematics, Polish Academy of Sciences,
Chopina 12, 87100 Torun, Poland, e-mail:trychlik@impan.gov.pl

(Received May 31, 1999; revised June 8, 2000)

Abstract.
We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single
observation. We also determine the distributions for which the bounds are attained.
The proof is based on combining the Moriguti monotone approximation of functions with the Hölder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.

Key words and phrases:
Order statistics, identically distributed variables, independent sample, dependent sample, $p$-th
central absolute moment, stability, sharp bound, Moriguti's monotone approximation, Hölder's inequality.

**Source**
(TeX , DVI )