IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v72y2020i3d10.1007_s10463-019-00709-3.html
   My bibliography  Save this article

Nonparametric estimation of the cross ratio function

Author

Listed:
  • Steven Abrams

    (Hasselt University
    University of Antwerp)

  • Paul Janssen

    (Hasselt University)

  • Jan Swanepoel

    (North-West University)

  • Noël Veraverbeke

    (Hasselt University
    North-West University)

Abstract

The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio, and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes. This dependence, however, disappears when the net income is large compared to the mean income.

Suggested Citation

  • Steven Abrams & Paul Janssen & Jan Swanepoel & Noël Veraverbeke, 2020. "Nonparametric estimation of the cross ratio function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 771-801, June.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:3:d:10.1007_s10463-019-00709-3
    DOI: 10.1007/s10463-019-00709-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10463-019-00709-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10463-019-00709-3?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. Alexandre Leblanc, 2012. "On estimating distribution functions using Bernstein polynomials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 919-943, October.
    2. Nan, Bin & Lin, Xihong & Lisabeth, Lynda D. & Harlow, Sioban D., 2006. "Piecewise Constant Cross-Ratio Estimation for Association of Age at a Marker Event and Age at Menopause," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 65-77, March.
    3. Janssen, Paul & Swanepoel, Jan & Veraverbeke, Noël, 2014. "A note on the asymptotic behavior of the Bernstein estimator of the copula density," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 480-487.
    4. Ingrid Van Keilegom & Noël Veraverbeke, 2001. "Hazard Rate Estimation in Nonparametric Regression with Censored Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 730-745, December.
    5. Bouezmarni Taoufik & Ghouch El & Taamouti Abderrahim, 2013. "Bernstein estimator for unbounded copula densities," Statistics & Risk Modeling, De Gruyter, vol. 30(4), pages 343-360, December.
    6. Li, Yi & Lin, Xihong, 2006. "Semiparametric Normal Transformation Models for Spatially Correlated Survival Data," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 591-603, June.
    7. Tianle Hu & Bin Nan & Xihong Lin & James M. Robins, 2011. "Time-dependent cross ratio estimation for bivariate failure times," Biometrika, Biometrika Trust, vol. 98(2), pages 341-354.
    8. Jan W. H. Swanepoel & Francois C. Van Graan, 2005. "A New Kernel Distribution Function Estimator Based on a Non‐parametric Transformation of the Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 551-562, December.
    9. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(3), pages 535-562, June.
    10. Janssen, Paul & Swanepoel, Jan & Veraverbeke, Noël, 2017. "Smooth copula-based estimation of the conditional density function with a single covariate," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 39-48.
    11. Luc Duchateau & Paul Janssen & Iva Kezic & Catherine Fortpied, 2003. "Evolution of recurrent asthma event rate over time in frailty models," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 52(3), pages 355-363, July.
    12. Spierdijk, Laura, 2008. "Nonparametric conditional hazard rate estimation: A local linear approach," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2419-2434, January.
    13. David V. Glidden, 2007. "Pairwise dependence diagnostics for clustered failure-time data," Biometrika, Biometrika Trust, vol. 94(2), pages 371-385.
    14. Yi Li & Ross L. Prentice & Xihong Lin, 2008. "Semiparametric maximum likelihood estimation in normal transformation models for bivariate survival data," Biometrika, Biometrika Trust, vol. 95(4), pages 947-960.
    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. Moshe Kelner & Zinoviy Landsman & Udi E. Makov, 2021. "Compound Archimedean Copulas," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(3), pages 126-126, June.

    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. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    2. Serge B. Provost & Yishan Zang, 2024. "Nonparametric Copula Density Estimation Methodologies," Mathematics, MDPI, vol. 12(3), pages 1-35, January.
    3. Janssen, Paul & Swanepoel, Jan & Veraverbeke, Noël, 2017. "Smooth copula-based estimation of the conditional density function with a single covariate," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 39-48.
    4. Plischke, Elmar & Borgonovo, Emanuele, 2019. "Copula theory and probabilistic sensitivity analysis: Is there a connection?," European Journal of Operational Research, Elsevier, vol. 277(3), pages 1046-1059.
    5. Eddie Anderson & Artem Prokhorov & Yajing Zhu, 2020. "A Simple Estimator of Two‐Dimensional Copulas, with Applications," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(6), pages 1375-1412, December.
    6. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Papers 2011.00909, arXiv.org, revised Mar 2021.
    7. Lajmi Lakhal-Chaieb & Thierry Duchesne, 2017. "Association measures for bivariate failure times in the presence of a cure fraction," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(4), pages 517-532, October.
    8. Nagler Thomas & Schellhase Christian & Czado Claudia, 2017. "Nonparametric estimation of simplified vine copula models: comparison of methods," Dependence Modeling, De Gruyter, vol. 5(1), pages 99-120, January.
    9. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
    10. Jing Ning & Karen Bandeen-Roche, 2014. "Estimation of time-dependent association for bivariate failure times in the presence of a competing risk," Biometrics, The International Biometric Society, vol. 70(1), pages 10-20, March.
    11. Mirza Nazmul Hasan & Roel Braekers, 2021. "Estimation of the association parameters in hierarchically clustered survival data by nested Archimedean copula functions," Computational Statistics, Springer, vol. 36(4), pages 2755-2787, December.
    12. Gneyou, Kossi Essona, 2014. "A strong linear representation for the maximum conditional hazard rate estimator in survival analysis," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 10-18.
    13. D. Blanke & D. Bosq, 2018. "Polygonal smoothing of the empirical distribution function," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 263-287, July.
    14. Lu Lu & Sujit Ghosh, 2023. "Nonparametric Estimation of Multivariate Copula Using Empirical Bayes Methods," Mathematics, MDPI, vol. 11(20), pages 1-22, October.
    15. Shahid Latif & Slobodan P. Simonovic, 2022. "Nonparametric Approach to Copula Estimation in Compounding The Joint Impact of Storm Surge and Rainfall Events in Coastal Flood Analysis," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 36(14), pages 5599-5632, November.
    16. Cheng, Guang & Zhou, Lan & Chen, Xiaohong & Huang, Jianhua Z., 2014. "Efficient estimation of semiparametric copula models for bivariate survival data," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 330-344.
    17. Ruosha Li & Yu Cheng & Qingxia Chen & Jason Fine, 2017. "Quantile association for bivariate survival data," Biometrics, The International Biometric Society, vol. 73(2), pages 506-516, June.
    18. Tianle Hu & Bin Nan & Xihong Lin, 2019. "Proportional cross-ratio model," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 25(3), pages 480-506, July.
    19. Aurélie Bertrand & Ingrid Van Keilegom & Catherine Legrand, 2019. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," Biometrics, The International Biometric Society, vol. 75(1), pages 297-307, March.
    20. Tian Dai & Ying Guo & Limin Peng & Amita K. Manatunga, 2018. "A local agreement pattern measure based on hazard functions for survival outcomes," Biometrics, The International Biometric Society, vol. 74(1), pages 86-99, March.

    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:spr:aistmt:v:72:y:2020:i:3:d:10.1007_s10463-019-00709-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.