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On the Gaussian Mixture Representation of the Laplace Distribution

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  • Peng Ding
  • Joseph K. Blitzstein

Abstract

Under certain conditions, a symmetric unimodal continuous random variable ξ can be represented as a scale mixture of a standard Normal distribution Z, that is, ξ=WZ$\xi = \sqrt{W} Z$, where the mixing distribution W is independent of Z. It is well known that if the mixing distribution is inverse Gamma, then ξ has Student’s t distribution. However, it is less well known that if the mixing distribution is Gamma, then ξ has a Laplace distribution. Several existing proofs of the latter result rely on complex calculus or nontrivial change of variables in integrals. We offer two simple and intuitive proofs based on representation and moment generating functions. As a byproduct, our proof by representation makes connections to many existing results in statistics. Supplementary materials for this article are available online.

Suggested Citation

  • Peng Ding & Joseph K. Blitzstein, 2018. "On the Gaussian Mixture Representation of the Laplace Distribution," The American Statistician, Taylor & Francis Journals, vol. 72(2), pages 172-174, April.
    • Handle: RePEc:taf:amstat:v:72:y:2018:i:2:p:172-174
    DOI: 10.1080/00031305.2017.1291448
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    References listed on IDEAS

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    1. Park, Trevor & Casella, George, 2008. "The Bayesian Lasso," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 681-686, June.
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    Cited by:

    1. Kozubowski, Tomasz J. & Mazur, Stepan & Podgorski, Krysztof, 2022. "Matrix Variate Generalized Laplace Distributions," Working Papers 2022:7, Örebro University, School of Business.

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