On the calculus of the cross-moments between an order statistic and a non ordered sample

In this paper we propose a method to calculate the joint cumulative distribution function (cdf) between any arbitrary order statistic and non-ordered samples of an N-compound sequence. Also, we show the need to compute their cross-moments by using the joint cdf, instead of the joint probability density function (pdf) as it is usual. This fact is caused by the non differentiability of the joint cdf. However, by using the knowledge of the cdf, we have found an expression too much complicated for most of the cases. So that, we suggest an alternative way of computing them like a linear combination of cross-moments between order statistics.