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Exponential and Hypoexponential Distributions: Some Characterizations

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  • George P. Yanev

    (Center for Vector-Borne Diseases, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
    Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria)

Abstract

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n ≥ 2 , X 1 , X 2 , … , X n are independent copies of a random variable X with unknown distribution F and a specific linear combination of X j ’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

Suggested Citation

  • George P. Yanev, 2020. "Exponential and Hypoexponential Distributions: Some Characterizations," Mathematics, MDPI, vol. 8(12), pages 1-10, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2207-:d:461108
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    References listed on IDEAS

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    1. Arnold, Barry C. & Villasenor, Jose A., 2013. "Exponential characterizations motivated by the structure of order statistics in samples of size two," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 596-601.
    2. Kim-Hung Li & Cheuk Ting Li, 2019. "Linear Combination of Independent Exponential Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 253-277, March.
    3. Sen, Ananda & Balakrishnan, N., 1999. "Convolution of geometrics and a reliability problem," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 421-426, July.
    4. Khaled Smaili & Therrar Kadri & Seifedine Kadry, 2016. "Finding the PDF of the hypoexponential random variable using the Kad matrix similar to the general Vandermonde matrix," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(5), pages 1542-1549, March.
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    Cited by:

    1. Schulz, Jan & Mayerhoffer, Daniel M., 2021. "A network approach to consumption," BERG Working Paper Series 173, Bamberg University, Bamberg Economic Research Group.

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