IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v190y2022ics016771522200147x.html
   My bibliography  Save this article

Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution

Author

Listed:
  • Soltani, A.R.

Abstract

In this article, firstly, we prove that functions of the form ϕ(x)=ecxI(−∞,0)(x)+ebxI[0,+∞)(x), c,b constants, are the only solutions to the integral equation ϕ(x)=∫01ϕ(ux)ϕ((1−u)x)du. This indeed gives the result of Van Asshe (1987) who used the Schwartz distribution theory to prove that for i.i.d X and Y, UX+(1−U)Y=dX if and only if X has a Cauchy distribution. Secondly, by looking into certain recursive integral equations involving characteristic functions, we prove that if for an n≥2, the random weight mean U(1)X1+(U(2)−U(1))X2+⋯+(1−U(n−1))Xn has a Cauchy distribution, then X1 has a Cauchy distribution; random variables X1,…,Xn are i.i.d, the random weights are the cuts of (0,1) by a uniform sample. The multivariate analogue of this result is also provided.

Suggested Citation

  • Soltani, A.R., 2022. "Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution," Statistics & Probability Letters, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:stapro:v:190:y:2022:i:c:s016771522200147x
    DOI: 10.1016/j.spl.2022.109606
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S016771522200147X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2022.109606?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Soltani, Ahmad Reza & Roozegar, Rasool, 2012. "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1012-1020.
    2. Soltani, A.R. & Homei, H., 2009. "Weighted averages with random proportions that are jointly uniformly distributed over the unit simplex," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1215-1218, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Soltani, Ahmad Reza & Roozegar, Rasool, 2012. "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1012-1020.
    2. Hazhir Homei, 2015. "A novel extension of randomly weighted averages," Statistical Papers, Springer, vol. 56(4), pages 933-946, November.
    3. Homei, H., 2012. "Randomly weighted averages with beta random proportions," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1515-1520.
    4. Roozegar, Rasool & zarch, Hamid Reza Taherizadeh, 2021. "On the asymptotic distribution of randomly weighted averages of random vectors," Statistics & Probability Letters, Elsevier, vol. 179(C).
    5. Ahmad Soltani & Kamel Abdollahnezhad, 2012. "On adjusted method of moments estimators on uniform distribution samples," METRON, Springer;Sapienza Università di Roma, vol. 70(1), pages 27-40, April.
    6. Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.
    7. Julian Górny & Erhard Cramer, 2019. "From B-spline representations to gamma representations in hybrid censoring," Statistical Papers, Springer, vol. 60(4), pages 1119-1135, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:190:y:2022:i:c:s016771522200147x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.