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On discrete α-unimodality

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Abstract

A continuous composition semigroup of probability generating functionsF≔(F t ,t≥0) and the corresponding multiplication ⊙ F of van Harnet al. (1982,Z. Wahrsch. Verw. Gebiete,61, 97–118) are used to introduce the concept of [F; α]-unimodality which generalizes the discrete α-unimodality due to Abouammoh (1987,Statist. Neerlandica,41, 239–244) and Alamatsaz (1993,Statist. Neerlandica,47, 245–252). We offer various characterizations and other properties of [F;α]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between [F;α]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of [F;α]-monotone and [F;α]-unimodal random variables are derived and some examples are also mentioned.

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Authors and Affiliations

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

    Emad-Eldin A. A. Aly

  2. Department of Mathematics, University of Indianapolis, 46227, Indianapolis, IN, U.S.A.

    Nadjib Bouzar

Authors
  1. Emad-Eldin A. A. Aly
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  2. Nadjib Bouzar
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Additional information

Research supported by Grant SS024 of the Research Center of Kuwait University.

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Aly, EE.A.A., Bouzar, N. On discrete α-unimodality. Ann Inst Stat Math 55, 523–535 (2003). https://doi.org/10.1007/BF02517805

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  • DOI: https://doi.org/10.1007/BF02517805

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