Characterizations Based on Regression Assumptions of Order Statistics

Let X-(1)<= X-(2)<=center dot center dot center dot <= X-(n) be the order statistics from independent and identically distributed random variables {X-i, 1 <= i <= n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.