A useful property of order statistics and its application to UMVU estimation

Consider a random sample of sizen drawn from a continuous parent distributionF. A basic and useful known property associated with such sample is the following: the conditional distribution of thej ^th order statistic given a valuet of thei ^th order statistics, (j>i), coincides with the distribution of the(j?i) ^th order statistic in a sample of size (n?i) drawn from the parent distributionF truncated at the left att. In this article we mention some applications of this property, and provide a new application to the construction of an Uniformly Minimum Variance Unbiased (UMVU) estimator in the case of two-truncation parameters family of distributions.