On characterizing distributions via linearity of regression for order statistics

Let X-1,...,X-n be a random sample from an absolutely continuous distribution with the corresponding order statistics X-1:n less than or equal to X-2:n less than or equal to X-n:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of Xk+2:n with respect to X-k:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.