# Order statistics from overlapping samples: bivariate densities and regression properties

## TL;DR

This paper derives explicit joint distributions and regression properties of order statistics from overlapping samples, providing new characterizations and insights into parent distribution identification.

## Contribution

It introduces explicit formulas for joint distributions of overlapping order statistics and explores regression-based characterizations of the parent distribution.

## Key findings

- Explicit formulas for joint distributions of overlapping order statistics.
- New regression characterizations for parent distribution identification.
- Results on uniqueness and identifiability of distributions from regressions.

## 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 (with respect to the Lebesgue measure on the real line). The distribution is identified through the form of the density with respect to a measure which is a sum of the bivariate Lebesgue measure on $\R^2$ and the univariate Lebesgue measure on the diagonal $\{(x,x):\,x\in\R\}$. 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. Nevertheless the general question of identifiability of the parent distribution by regression of order statistics from overlapping samples remains open.