A CHARACTERIZATION OF DISCRETE UNIMODALITY WITH APPLICATIONS TO VARIANCE UPPER BOUNDS

SHARON E. NAVARD^{1}, JOHN W. SEAMAN, JR.^{2} AND DEAN M. YOUNG^{2}

1Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, VA 23284-2014, U.S.A.

2Department of Information Systems, Baylor University, Waco, TX 76798-8005, U.S.A.

(Received June 4, 1990; revised January 14, 1992)

Abstract. Bertin and Theodorescu (1984, Statist. Probab. Lett., 2, 23–30) developed a characterization of discrete unimodality based on convexity properties of a discretization of distribution functions. We offer a new characterization of discrete unimodality based on convexity properties of a piecewise linear extension of distribution functions. This reliance on functional convexity, as in Khintchine’s classic definition, leads to variance dilations and upper bounds on variance for a large class of discrete unimodal distributions. These bounds are compared to existing inequalities due to Muilwijk (1966, Sankhyā, Ser. B, 28, p. 183), Moors and Muilwijk (1971, Sankhyā, Ser. B, 33, 385–388), and Rayner (1975, Sankhyā, Ser. B, 37, 135–138), and are found to be generally tighter, thus illustrating the power of unimodality assumptions.

Key words and phrases: Discrete distributions, unimodality, convexity, variance bounds.