METRON

, Volume 74, Issue 1, pp 75–97 | Cite as

Nonparametric relative regression for associated random variables

Article
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Abstract

We consider the problem of estimating the regression function based on the minimization of the mean squared relative error. We construct an alternative kernel estimate of the regression function. We prove the strong consistency and the asymptotic normality of the constructed estimator under weak dependence conditions. We conduct a simulation study to compare the finite sample performance of our estimator with that of the classical kernel regression.

Keywords

Kernel method Relative error Non-parametric estimation  Associated variable  

Mathematics Subject Classification

62G20 62G08 

References

  1. 1.
    Barlow, R.E., Proschan, F.: Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart & Winston, New York (1975)MATHGoogle Scholar
  2. 2.
    Bulinski, A., Suquet, C.: Normal approximation for quasi-associated random fields. Stat. Probab. Lett. 54, 215–226 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chatfield, C.: The joys of consulting. Significance 4, 3336 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, K., Guo, S., Lin, Y., Ying, Z.: Least absolute relative error estimation. J. Am. Stat. Assoc. 105, 1104–1112 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Douge, L.: Thormes limites pour des variables quasi-associes hilbertiennes. Ann. I.S.U.P. 54, 51–60 (2010)MathSciNetGoogle Scholar
  6. 6.
    Esary, J., Proschan, F., Walkup, D.: Association of random variables with applications. Ann. Math. Stat. 38, 1466–1476 (1967)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gramaglia, M., Trullols-Cruces, O., Naboulsi, D., Fiore, M., Calderon, M.: Vehicular Networks on Two Madrid Highways, [Technical Report] (2014). https://hal.inria.fr/hal-00959837v1
  8. 8.
    Jones, M.C., Park, H., Shin, K.L., Vines, S.K., Jeong, S.O.: Relative error prediction via kernel regression smoothers. J. Stat. Plan. Inference 138, 2887–2898 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jong-Dev, K., Proschan, F.: Negative association of random variables, with applications. Ann. Stat. 11, 286–295 (1983)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kallabis, R.S., Neumann, M.H.: An exponential inequality under weak dependence. Bernoulli 12, 333–350 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khoshgoftaar, T.M., Bhattacharyya, B.B., Richardson, G.D.: Predicting software errors, during development, using nonlinear regression models: a comparative study. IEEE Trans. Reliab. 41, 390–395 (1992)CrossRefMATHGoogle Scholar
  12. 12.
    Lin, Z., Li, D.: Asymptotic normality for \(L_1\)-norm kernel estimator of conditional median under association dependence. J. Multivar. Anal. 98, 1214–1230 (2007)CrossRefMATHGoogle Scholar
  13. 13.
    Narula, S.C., Wellington, J.F.: Prediction, linear regression and the minimum sum of relative errors. Technometrics 19, 185190 (1977)CrossRefMATHGoogle Scholar
  14. 14.
    Shen, V.Y., Yu, T., Thebaut, S.M.: Identifying error-prone softwarean empirical study. IEEE Trans. Softw. Eng. 11, 317324 (1985)Google Scholar
  15. 15.
    Wilcox, R.: Introduction to robust estimation and hypothesis testing. Academic Press, Cambridge (2005)MATHGoogle Scholar
  16. 16.
    Yang, Y., Ye, F.: General relative error criterion and M-estimation. Front. Math. China 8, 695–715 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Sapienza Università di Roma 2016

Authors and Affiliations

  1. 1.Laboratoire de statistique et processus stochastiquesUniv. Djillali LiabèsSidi Bel AbbèsAlgeria

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