A CHARACTERIZATION OF CERTAIN DISCRETE EXPONENTIAL FAMILIES

T. T. NGUYEN, A. K. GUPTA AND Y. WANG

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403-0221, U.S.A.

(Received February 22, 1995; revised August 2, 1995)

Abstract. For independent random variables X and Y, define S ≡ X + Y. When the conditional expectations E[g(X) | S] ≡ a(S) and E[h(X) | S] ≡ b(S) are given, then under certain assumptions, the density function of X has the form of u(x)k(α)e^{αx}, where u(x) is uniquely determined by the functions a(·) and b(·).

Key words and phrases: Characterization, conditional expectation.