Note on characterization problem of Nagaraja and Nevzorov

Abstract

In this note a conjecture of Nagaraja and Nevzorov (1997) concerning the characterization of distributions functions by a convex conditional mean function is proved. It is not neccesary furthermore to suppose that the c.d. f. is continuous and the characterization does hold without assumptions considered in the result of the above reference. The present note gives a general theorem of characterization in terms of the convex conditional mean function introduced by Nagaraja and Nevzorov (1997) determining explicitly the distribution function associated with each one.