Upper non positive bounds on expectations of generalized order statistics from DD and DDA populations

ABSTRACT

We present the upper non positive bounds on the expectations of gOSs centered about the sample mean, which are based on the parent distributions with decreasing density and decreasing density on average distributions. Such bounds can be obtained only for particular cases of gOSs and they are expressed in units generated by the central absolute moments of a fixed order. The attainability conditions are also described. The method of deriving presented bounds is based on the maximization of appropriate norms over properly chosen convex sets. The paper complements the results of Bieniek [J. Statist. Plann. Inference, 2008; 138:971–981].

KEYWORDS:

• Bound
• convex transform order
• decreasing density on the average
• decreasing density
• generalized order statistics.

MATHEMATICS SUBJECT CLASSIFICATION:

• 60E15
• 62G32

Acknowledgments

The author is very grateful to Tomasz Rychlik from the Polish Academy of Sciences, and to the anonymous referee, whose very helpful comments and valuable remarks helped to significantly improve the paper.

Funding

The research was supported by the Polish National Science Center [grant number 2011/01/D/ST1/04172].