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Simple proofs of two results on convolutions of unimodal distributions

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  • Purkayastha, Sumitra

Abstract

We give simple proofs of two results about convolutions of unimodal distributions. The first of these results states that the convolution of two symmetric unimodal distributions on is unimodal. The other result states that symmetrization of a unimodal random variable gives a symmetric unimodal random variable. Both our proofs avoid Khintchine's representation of a random variable that is unimodal about zero, and use the integral representation of the expectation of a non-negative random variable with its tail probability as the integrand.

Suggested Citation

  • Purkayastha, Sumitra, 1998. "Simple proofs of two results on convolutions of unimodal distributions," Statistics & Probability Letters, Elsevier, vol. 39(2), pages 97-100, August.
    • Handle: RePEc:eee:stapro:v:39:y:1998:i:2:p:97-100
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    Cited by:

    1. Mahdi Alimohammadi & Mohammad Hossein Alamatsaz & Erhard Cramer, 2016. "Convolutions and generalization of logconcavity: Implications and applications," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(2), pages 109-123, March.
    2. Tianqing Liu & Xiaohui Yuan, 2020. "Empirical likelihood-based weighted rank regression with missing covariates," Statistical Papers, Springer, vol. 61(2), pages 697-725, April.
    3. Steiner, Jakub & Netzer, Nick & Robson, Arthur & Kocourek, Pavel, 2021. "Endogenous Risk Attitudes," CEPR Discussion Papers 16190, C.E.P.R. Discussion Papers.
    4. Ali E. Abbas & James E. Matheson & Robert F. Bordley, 2009. "Effective utility functions induced by organizational target-based incentives," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 30(4), pages 235-251.

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