CHARACTERIZATIONS BASED ON CONDITIONAL EXPECTATIONS OF THE DOUBLED TRUNCATED DISTRIBUTION

J. M. RUIZ AND J. NAVARRO

Departamento de Matemática Aplicada y Estadística, Universidad de Murcia, 30100 Murcia, Spain

(Received February 28, 1994; revised September 21, 1995)

Abstract. The expression of the continuous distribution function F (x) is obtained whenever m(x, y) = E(X | x ≤ X ≤ y) is known. Moreover, we obtain the necessary and sufficient conditions so that any function m : R^{2} → R is the conditional expectation E(X | x ≤ X ≤ y) of a random variable X with continuous distribution function. Furthermore, we relate m(x, y) to order statistics.

Key words and phrases: Characterizations, doubled truncated distribution, conditional expectations, mean residual life, order statistics.