Characterizations based on certain regression assumptions of adjacent order statistics

Let X(1)≤X(2)≤⋯≤X(n)X(1)≤X(2)≤⋯≤X(n) be the order statistics derived from independent and identically distributed random variables {Xi,1≤i≤n}{Xi,1≤i≤n} with a common absolutely continuous distribution function. We investigate characterizations of distributions by using the equality and linearity of E(X(1)2−(ηX(2)+θ)X(1)|X(2)) and E(X(n)2−(ηX(n−1)+θ)X(n)|X(n−1)), where ηη and θθ are constants. It turns out a large class of distributions can be characterized. In particular, many important distributions, such as the normal, gamma, exponential, inverse gamma, Student tt, and uniform distributions, can be characterized correspondingly. Similar characterizations by using analogous regressional properties within the class of sample processes can also be obtained.