A NOTION OF α-MONOTONICITY WITH GENERALIZED MULTIPLICATIONS

EMAD-ELDIN A. A. ALY^{1} AND NADJIB BOUZAR^{2}

1Department of Statistics and O.R., Kuwait University, P.O.B. 5969, Safat 13060, Kuwait,

e-mail: emad@kuc01.kuniv.edu.kw

2Department of Mathematics, University of Indianapolis, Indianapolis, IN 46227, U.S.A., e-mail: nbouzar@uindy.edu

(Received February 1, 2000; revised August 24, 2000)

Abstract. The multiplications of van Harn et al. (1982, Z. Wahrsch. Verw. Gebiete, 61, 97–118) are used to generalize the definition of α-monotonicity of Olshen and Savage (1970, J. Appl. Probab., 7, 21–34) and Steutel (1988, Statist. Neerlandica, 42, 137–140) for distributions with support in Z Z Z Z Z Z Z Z_{+} and R R R R R R R R_{+}. Several characterizations are offered and a convolution property is established. Some relevant stability equations are solved and a relationship with the important concept of self-decomposability is noted. Poisson mixtures are used to deduce results for the R R R R R R R R_{+}-case from those for the Z Z Z Z Z Z Z Z_{+}-case.

Key words and phrases: Semigroup, monotonicity, self-decomposability, Poisson mixtures.