IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v146y2016icp261-268.html
   My bibliography  Save this article

Relative-error prediction in nonparametric functional statistics: Theory and practice

Author

Listed:
  • Demongeot, Jacques
  • Hamie, Ali
  • Laksaci, Ali
  • Rachdi, Mustapha

Abstract

In this paper, an alternative kernel estimator of the regression operator of a scalar response variable Y given a random variable X taking values in a semi-metric space is considered. The constructed estimator is based on the minimization of the mean squared relative error. This technique is useful in analyzing data with positive responses, such as stock prices or life times. Least squares or least absolute deviation are among the most widely used criteria in statistical estimation for regression models. However, in many practical applications, especially in treating, for example, the stock price data, the size of the relative error rather than that of the error itself, is the central concern of the practitioners. This paper offers then an alternative to traditional estimation methods by considering the minimization of the least absolute relative error for operatorial regression models. We prove the strong and the uniform consistencies (with rates) of the constructed estimator. Moreover, the mean squared convergence rate is given and the asymptotic normality of the proposed estimator is proved. Finally, supportive evidence is shown by simulation studies and an application on some economic data was performed.

Suggested Citation

  • Demongeot, Jacques & Hamie, Ali & Laksaci, Ali & Rachdi, Mustapha, 2016. "Relative-error prediction in nonparametric functional statistics: Theory and practice," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 261-268.
    • Handle: RePEc:eee:jmvana:v:146:y:2016:i:c:p:261-268
    DOI: 10.1016/j.jmva.2015.09.019
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X1500233X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2015.09.019?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. Masry, Elias, 2005. "Nonparametric regression estimation for dependent functional data: asymptotic normality," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 155-177, January.
    2. K. Benhenni & F. Ferraty & M. Rachdi & P. Vieu, 2007. "Local smoothing regression with functional data," Computational Statistics, Springer, vol. 22(3), pages 353-369, September.
    3. Kudraszow, Nadia L. & Vieu, Philippe, 2013. "Uniform consistency of kNN regressors for functional variables," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1863-1870.
    4. Park, Heungsun & Stefanski, L. A., 1998. "Relative-error prediction," Statistics & Probability Letters, Elsevier, vol. 40(3), pages 227-236, October.
    5. J. Barrientos-Marin & F. Ferraty & P. Vieu, 2010. "Locally modelled regression and functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(5), pages 617-632.
    6. Amiri, Aboubacar & Crambes, Christophe & Thiam, Baba, 2014. "Recursive estimation of nonparametric regression with functional covariate," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 154-172.
    7. Chen, Kani & Guo, Shaojun & Lin, Yuanyuan & Ying, Zhiliang, 2010. "Least Absolute Relative Error Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1104-1112.
    8. Rachdi, Mustapha & Laksaci, Ali & Demongeot, Jacques & Abdali, Abdel & Madani, Fethi, 2014. "Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 53-68.
    9. Aldo Goia & Philippe Vieu, 2015. "A partitioned Single Functional Index Model," Computational Statistics, Springer, vol. 30(3), pages 673-692, September.
    10. Azzedine, Nadjia & Laksaci, Ali & Ould-Saïd, Elias, 2008. "On robust nonparametric regression estimation for a functional regressor," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3216-3221, December.
    11. C. Crambes & L. Delsol & A. Laksaci, 2008. "Robust nonparametric estimation for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(7), pages 573-598.
    12. Aneiros-Pérez, Germán & Vieu, Philippe, 2006. "Semi-functional partial linear regression," Statistics & Probability Letters, Elsevier, vol. 76(11), pages 1102-1110, June.
    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. Ouafae Benrabah & Feriel Bouhadjera & Elias Ould Saïd, 2022. "Local linear estimation of the regression function for twice censored data," Statistical Papers, Springer, vol. 63(2), pages 489-514, April.
    2. Slaoui, Yousri, 2019. "Wild bootstrap bandwidth selection of recursive nonparametric relative regression for independent functional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 494-511.
    3. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.
    4. Slaoui Yousri & Khardani Salah, 2020. "Nonparametric relative recursive regression," Dependence Modeling, De Gruyter, vol. 8(1), pages 221-238, January.
    5. Slaoui Yousri & Khardani Salah, 2020. "Nonparametric relative recursive regression," Dependence Modeling, De Gruyter, vol. 8(1), pages 221-238, January.

    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. Zhiyong Zhou & Zhengyan Lin, 2016. "Asymptotic normality of locally modelled regression estimator for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 116-131, March.
    2. Kara, Lydia-Zaitri & Laksaci, Ali & Rachdi, Mustapha & Vieu, Philippe, 2017. "Data-driven kNN estimation in nonparametric functional data analysis," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 176-188.
    3. Bouzebda, Salim & Chaouch, Mohamed, 2022. "Uniform limit theorems for a class of conditional Z-estimators when covariates are functions," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    4. Slaoui, Yousri, 2019. "Wild bootstrap bandwidth selection of recursive nonparametric relative regression for independent functional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 494-511.
    5. Chagny, Gaëlle & Roche, Angelina, 2016. "Adaptive estimation in the functional nonparametric regression model," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 105-118.
    6. Gheriballah, Abdelkader & Laksaci, Ali & Sekkal, Soumeya, 2013. "Nonparametric M-regression for functional ergodic data," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 902-908.
    7. Chaouch, Mohamed, 2019. "Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 129-148.
    8. Mustapha Rachdi & Ali Laksaci & Zoulikha Kaid & Abbassia Benchiha & Fahimah A. Al‐Awadhi, 2021. "k‐Nearest neighbors local linear regression for functional and missing data at random," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(1), pages 42-65, February.
    9. Lydia Kara-Zaitri & Ali Laksaci & Mustapha Rachdi & Philippe Vieu, 2017. "Uniform in bandwidth consistency for various kernel estimators involving functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(1), pages 85-107, January.
    10. Eduardo L. Montoya & Wendy Meiring, 2016. "An F-type test for detecting departure from monotonicity in a functional linear model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 322-337, June.
    11. Rachdi, Mustapha & Laksaci, Ali & Demongeot, Jacques & Abdali, Abdel & Madani, Fethi, 2014. "Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 53-68.
    12. Oussama Bouanani & Saâdia Rahmani & Ali Laksaci & Mustapha Rachdi, 2020. "Asymptotic normality of conditional mode estimation for functional dependent data," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(2), pages 465-481, June.
    13. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    14. Fahimah A. Al-Awadhi & Zoulikha Kaid & Ali Laksaci & Idir Ouassou & Mustapha Rachdi, 2019. "Functional data analysis: local linear estimation of the $$L_1$$ L 1 -conditional quantiles," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(2), pages 217-240, June.
    15. Han Shang, 2014. "Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density," Computational Statistics, Springer, vol. 29(3), pages 829-848, June.
    16. Shang, Han Lin, 2013. "Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 185-198.
    17. Benhenni, Karim & Hassan, Ali Hajj & Su, Yingcai, 2019. "Local polynomial estimation of regression operators from functional data with correlated errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 80-94.
    18. Karim Benhenni & Sonia Hedli-Griche & Mustapha Rachdi, 2017. "Regression models with correlated errors based on functional random design," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 1-21, March.
    19. Han Lin Shang, 2014. "Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(3), pages 599-615, September.
    20. Boente, Graciela & Vahnovan, Alejandra, 2017. "Robust estimators in semi-functional partial linear regression models," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 59-84.

    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:jmvana:v:146:y:2016:i:c:p:261-268. 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.