  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.
