Journal of Systems Science and Complexity

, Volume 30, Issue 5, pp 1160–1172 | Cite as

A relative error estimation approach for multiplicative single index model

  • Zhanfeng WangEmail author
  • Zimu Chen
  • Yaohua Wu


As an alternative to absolute error methods, such as the least square and least absolute deviation estimations, a product relative error estimation is proposed for a multiplicative single index regression model. Regression coefficients in the model are estimated via a two-stage procedure and their statistical properties such as consistency and normality are studied. Numerical studies including simulation and a body fat example show that the proposed method performs well.


Asymptotic properties least product relative error relative errors single index model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Chen K N, Guo S J, Lin Y Y, et al., Least absolute relative error estimation, Journal of the American Statistical Association, 2010, 105(491): 1104–1112.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chen K N, Lin Y Y, Wang Z F, et al., Least product relative error estimation, Journal of Multivariate Analysis, 2016, 144: 91–98.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Park H S and Stefanski L A, Relative-error prediction, Statistics & Probability Letters, 1998, 40(3): 227–236.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Ye J M, Price models and the value relevance of accounting information, SSRN Electronic Journal,, 2007.Google Scholar
  5. [5]
    Zhang Q Z and Wang Q H, Local least absolute relative error estimating approach for partially linear multiplicative model, Statistica Sinica, 2013, 23(3): 1091–1116.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Wang Z F, Liu W X, and Lin Y Y, A change-point problem in relative error-based regression, TEST, 2015, 24(4): 835–856.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Hristache M, Juditsky A, and Spokoiny V, Direct estimation of the index coefficient in a singleindex model, Annals of Statistics, 2001, 29(3): 595–623.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Ruppert D, Wand M P, and Carroll R J, Semiparametric Regression, Cambridge University Press, London, 2003.CrossRefzbMATHGoogle Scholar
  9. [9]
    Stute W and Zhu L X, Nonparametric checks for single-index models, The Annals of Statistics, 2005, 33(3): 1048–1083.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Zhu L X and Xue L G, Empirical likelihood confidence regions in a partially linear single-index model, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2006, 68(3): 549–570.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Wang J L, Xue L G, Zhu L X, et al., Estimation for a partial linear single-index model, The Annals of Statistics, 2010, 38(1): 246–274.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Chang Z Q, Xue L G, and Zhu L X, On an asymptotically more efficient estimation of the single-index model, Journal of Multivariate Analysis, 2010, 101(8): 1898–1901.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Kong E and Xia Y C, A single-index quantile regression model and its estimation, Econometric Theory, 2012, 28(4): 730–768.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Härdle W K, Müller M, Sperlich S, et al., Nonparametric and Semiparametric Models, Springer Science & Business Media, Berlin, 2012.zbMATHGoogle Scholar
  15. [15]
    Fan J Q and Gijbels I, Local Polynomial Modelling and Its Applications: Monographs on Statistics and Applied Probability, CRC Press, London, 1996.zbMATHGoogle Scholar
  16. [16]
    Carroll R J, Fan J Q, Gijbels I, et al., Generalized partially linear single-index models, Journal of the American Statistical Association, 1997, 92(438): 477–489.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Yu Y and Ruppert D, Penalized spline estimation for partially linear single-index models, Journal of the American Statistical Association, 2002, 97(460): 1042–1054.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Li K C, Sliced inverse regression for dimension reduction, Journal of the American Statistical Association, 1991, 86(414): 316–327.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Xia Y C, Tong H, Li W K, et al., An adaptive estimation of dimension reduction space, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2002, 64(3): 363–410.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Li T T, Yang H, Wang J L, et al., Correction on estimation for a partial-linear single-index model, The Annals of Statistics, 2011, 39(6): 3441–3443.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Statistics and Finance, School of ManagementUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations