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On the gamma difference distribution

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  • Forrester, Peter J.

Abstract

The gamma difference distribution is defined as the difference of two independent gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear differential equation specification of the probability density function is given. This is used to derive a Stein-type differential identity relating to the expectation with respect to the gamma difference distribution of a general twice differentiable function g(x). Choosing g(x)=xk gives a second order recurrence for the positive integer moments, which are also shown to permit evaluations in terms of 2F1 hypergeometric polynomials. A hypergeometric function evaluation is given for the absolute continuous moments. Specialising the gamma difference distribution gives the variance gamma distribution. Results of the type obtained herein have previously been obtained for this distribution, allowing for comparisons to be made.

Suggested Citation

  • Forrester, Peter J., 2024. "On the gamma difference distribution," Statistics & Probability Letters, Elsevier, vol. 211(C).
    • Handle: RePEc:eee:stapro:v:211:y:2024:i:c:s0167715224001056
    DOI: 10.1016/j.spl.2024.110136
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    References listed on IDEAS

    as
    1. Gaunt, Robert E., 2023. "On the moments of the variance-gamma distribution," Statistics & Probability Letters, Elsevier, vol. 201(C).
    2. Robert E. Gaunt, 2021. "A simple proof of the characteristic function of Student’s t-distribution," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(14), pages 3380-3383, July.
    3. Shephard, N.G., 1991. "From Characteristic Function to Distribution Function: A Simple Framework for the Theory," Econometric Theory, Cambridge University Press, vol. 7(4), pages 519-529, December.
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