On the characterizing property of the convex conditional mean function

In this note, a conjecture of Nagaraja and Nevzorov (J. Statist. Plann. Inference 63 (1997) 271-284) concerning the characterization of a distribution function by its associated convex conditional mean function (CCMF) is proved. The characterizing property of the CCMF holds without the assumptions considered in the result of the above reference. As a consequence, we derive a result of characterization related to the uniform distribution. Applications to the characterization of a c.d.f. based on properties of certain regression functions associated with the order statistics are discussed.