On Characterizing Distributions via Linearity of Regression for Order Statistics

summary Let X 1 X n be a random sample from an absolutely continuous distribution with the corresponding order statistics X 1:n ≤ X 2:n ≤ X n:n . A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of X k+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.