Order statistics from overlapping samples: bivariate densities and regression properties

ABSTRACT

In this paper, we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample.

KEYWORDS:

• Characterization
• order statistics
• overlapping samples
• linearity of regression
• uniqueness theorem

MSC 2010 SUBJECT CLASSIFICATIONS:

• Primary: 60E05
• Secondary: 62E10

Acknowledgments

The authors are greatly indebted to the referees for valuable suggestions which helped significantly in improving the final version of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Nan-Cheng Su http://orcid.org/0000-0002-3629-3568

Additional information

Funding

Support for this research was provided in part by the Ministry of Science and Technology, Taiwan [grant numbers 102-2118-M-305-003 and 104-2118-M-305-003], by Ministerio de Economia, Industria y Competitividad, Spain [grant number MTM2016-74983] and by the National Science Center, Poland [grant number 2016/21/B/ST1/00005].