IDEAS home Printed from https://ideas.repec.org/a/gam/jstats/v5y2022i1p12-189d747049.html
   My bibliography  Save this article

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

Author

Listed:
  • Mei-Ling Ting Lee

    (School of Public Health, University of Maryland, College Park, MD 20742, USA)

  • George A. Whitmore

    (Faculty of Management, McGill University, Montreal, QC H3A 0G4, Canada
    Ottawa Hospital Research Institute, Ottawa, ON K1Y 4E9, Canada)

Abstract

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

Suggested Citation

  • Mei-Ling Ting Lee & George A. Whitmore, 2022. "Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property," Stats, MDPI, vol. 5(1), pages 1-18, February.
  • Handle: RePEc:gam:jstats:v:5:y:2022:i:1:p:12-189:d:747049
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2571-905X/5/1/12/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2571-905X/5/1/12/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Küchler, Uwe & Tappe, Stefan, 2008. "Bilateral gamma distributions and processes in financial mathematics," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 261-283, February.
    2. S. Sæbø & T. Almøy & A. H. Aastveit, 2005. "Disease resistance modelled as first‐passage times of genetically dependent stochastic processes," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(1), pages 273-285, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chrys Caroni, 2022. "Regression Models for Lifetime Data: An Overview," Stats, MDPI, vol. 5(4), pages 1-11, December.
    2. Roman V. Ivanov, 2023. "The Semi-Hyperbolic Distribution and Its Applications," Stats, MDPI, vol. 6(4), pages 1-21, October.

    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. Dilip B. Madan & King Wang, 2022. "Two sided efficient frontiers at multiple time horizons," Annals of Finance, Springer, vol. 18(3), pages 327-353, September.
    2. Pérez-Abreu, Victor & Stelzer, Robert, 2014. "Infinitely divisible multivariate and matrix Gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 155-175.
    3. Küchler, Uwe & Tappe, Stefan, 2013. "Tempered stable distributions and processes," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4256-4293.
    4. Xie, Haibin & Wang, Shouyang & Lu, Zudi, 2018. "The behavioral implications of the bilateral gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 259-264.
    5. Dilip B. Madan & Wim Schoutens & King Wang, 2020. "Bilateral multiple gamma returns: Their risks and rewards," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 1-27, March.
    6. Yoshihiro Shirai, 2023. "Acceptable Bilateral Gamma Parameters," Papers 2301.05333, arXiv.org.
    7. Küchler, Uwe & Tappe, Stefan, 2008. "On the shapes of bilateral Gamma densities," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2478-2484, October.
    8. Victor Korolev & Alexander Zeifman, 2023. "Quasi-Exponentiated Normal Distributions: Mixture Representations and Asymmetrization," Mathematics, MDPI, vol. 11(17), pages 1-14, September.
    9. Dilip B. Madan & Wim Schoutens, 2020. "Self‐similarity in long‐horizon returns," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1368-1391, October.
    10. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    11. Giorgia Callegaro & Lucio Fiorin & Martino Grasselli, 2019. "Quantization meets Fourier: a new technology for pricing options," Annals of Operations Research, Springer, vol. 282(1), pages 59-86, November.
    12. Dilip B. Madan & King Wang, 2023. "The valuation of corporations: a derivative pricing perspective," Annals of Finance, Springer, vol. 19(1), pages 1-21, March.
    13. Yoshihiro Shirai, 2022. "Extreme Measures in Continuous Time Conic Finace," Papers 2210.13671, arXiv.org, revised Oct 2023.
    14. Kais Hamza & Fima C. Klebaner & Zinoviy Landsman & Ying-Oon Tan, 2014. "Option Pricing for Symmetric L\'evy Returns with Applications," Papers 1402.1554, arXiv.org.
    15. Maha A. Omair & Yusra A. Tashkandy & Sameh Askar & Abdulhamid A. Alzaid, 2022. "Family of Distributions Derived from Whittaker Function," Mathematics, MDPI, vol. 10(7), pages 1-23, March.
    16. Tomy, Lishamol & Jose, K.K., 2009. "Generalized normal-Laplace AR process," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1615-1620, July.
    17. Ernst Eberlein & Antonis Papapantoleon & Albert Shiryaev, 2008. "On the duality principle in option pricing: semimartingale setting," Finance and Stochastics, Springer, vol. 12(2), pages 265-292, April.
    18. Lucio Fiorin & Wim Schoutens, 2020. "Conic quantization: stochastic volatility and market implied liquidity," Quantitative Finance, Taylor & Francis Journals, vol. 20(4), pages 531-542, April.
    19. Uehara, Yuma, 2019. "Statistical inference for misspecified ergodic Lévy driven stochastic differential equation models," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4051-4081.
    20. Schneider, Paul, 2015. "Generalized risk premia," Journal of Financial Economics, Elsevier, vol. 116(3), pages 487-504.

    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:gam:jstats:v:5:y:2022:i:1:p:12-189:d:747049. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.