A characterization of the rectangular distribution

It is assumed that the statistics S and T (given by formula (1.2) and (1.3)) have constant regression on Λ= ∑ⁿ_j=1X_j. Under cetain additonal assumptions this leads either to a symmetric two point distribution or the rectangular distribution over (?1, +1). The two point distribution can be excluded by assuming that the population distribution function is continuous. A characterization theorem for the rectangular distribution over (?1, +1), is then obtained.