Skip to main content
SpringerLink
Log in

Empirical likelihood inference for semi-parametric estimating equations

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Qin and Lawless (1994) established the statistical inference theory for the empirical likelihood of the general estimating equations. However, in many practical problems, some unknown functional parts h(t) appear in the corresponding estimating equations E F G(X, h(T), β) = 0. In this paper, the empirical likelihood inference of combining information about unknown parameters and distribution function through the semi-parametric estimating equations are developed, and the corresponding Wilk’s Theorem is established. The simulations of several useful models are conducted to compare the finite-sample performance of the proposed method and that of the normal approximation based method. An illustrated real example is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen J H, Qin J. Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika, 1993, 80: 107–116

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen S X, Qin Y S. Empirical likelihood confidence intervals for local linear smoothers. Biometrika, 2000, 87: 946–953

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen Q H, Zhong P S, Cui H J. Empirical likelihood for mixed-effects error-in-variables model. Acta Math Appl Sin, 2009, 25: 561–578

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen S X, Cui H J. On the second-order properties of empirical likelihood with moment restrictions. J Econometrics, 2007, 141: 492–516

    Article  MathSciNet  Google Scholar 

  5. Chen X, Cui H J. Empirical likelihood for partially linear single-index errors-in-variables model. Comm Statist Theory Methods, 2009, 38: 2498–2514

    Article  MathSciNet  MATH  Google Scholar 

  6. Chuang C S, Chan N H. Empirical likelihood for autoregressive models with applications to unstable time series. Statist Sinica, 2002, 12: 81–87

    MathSciNet  Google Scholar 

  7. Cui H J, Kong E F. Empirical likelihood confidence region for parameters in semi-linear errors-in-variables models. Scand J Statist, 2006, 33: 153–168

    Article  MathSciNet  MATH  Google Scholar 

  8. Kitamura Y C. Asymptotic optimality of empirical likelihood for testing moment restrictions. Econometrica, 2001, 69: 1661–1672

    Article  MathSciNet  MATH  Google Scholar 

  9. Kolaczyk E D. Empirical likelihood confidence for generalized linear models. Statist Sinica, 1994, 4: 199–218

    MathSciNet  MATH  Google Scholar 

  10. Li G, Wang Q H. Empirical likelihood regression analysis for right censored data. Statist Sinica, 2003, 13: 51–68

    MathSciNet  MATH  Google Scholar 

  11. Owen A B. Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 1988, 75: 237–249

    Article  MathSciNet  MATH  Google Scholar 

  12. Owen A B. Empirical likelihood ratio confidence regions. Ann Statist, 1990, 18: 90–120

    Article  MathSciNet  MATH  Google Scholar 

  13. Owen A B. Empirical likelihood for linear models. Ann Statist, 1991, 19: 1725–1747

    Article  MathSciNet  MATH  Google Scholar 

  14. Owen A B. Empirical likelihood and generalized projection pursuit. Technical report 393, Department of Statistics, Stanford University, 1992

  15. Qin J. Empirical likelihood in biased sample problems. Ann Statist, 1993, 21: 1182–1196

    Article  MathSciNet  MATH  Google Scholar 

  16. Qin J, Lawless J. Empirical likelihood and general estimating equations. Ann Statist, 1994, 22: 300–325

    Article  MathSciNet  MATH  Google Scholar 

  17. Qin Y S. Empirical likelihood ratio confidence regions in a partially linear model. Chinese J Appl Probab Statist, 1999, 15: 363–369

    MathSciNet  MATH  Google Scholar 

  18. Qin Y S, Lei Q Z. On empirical likelihood for linear models with missing responses. J Statist Plann Inference, 2010, 140: 3399–3408

    Article  MathSciNet  MATH  Google Scholar 

  19. Qin Y S, Li J J. Empirical likelihood for partially linear models with missing responses at random. J Nonparametr Stat, 2011, 23: 497–511

    Article  MathSciNet  MATH  Google Scholar 

  20. Qin Y S, Rao J N K, Ren Q. Confidence intervals for marginal parameters under fractional linear regression imputation for missing data. J Multivariate Anal, 2008, 99: 1232–1259

    Article  MathSciNet  MATH  Google Scholar 

  21. Shi J, Lau T S. Empirical likelihood for partially linear models. J Multivariate Anal, 2000, 72: 132–148

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang Q H, Jing B Y. Empirical likelihood for partially linear models with fixed design. Statist Probab Lett, 1999, 41: 425–433

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang Q H, Jing B Y. Empirical likelihood for a class of functionals of survival distribution with censored data. Ann Inst Statist Math, 2001, 53: 517–527

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang Q H, Li G. Empirical likelihood semiparametric regression analysis under random censorship. J Multivariate Anal, 2002, 83: 469–486

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang Q H, Rao J N K. Empirical likelihood for linear regression models under imputation for missing responses. Canad J Statist, 2001, 29: 597–608

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang Q H, Rao J N K. Empirical likelihood-based inference in linear errors-in-covariables models with validation data. Biometrika, 2002, 89: 345–358

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang Q H, Rao J N K. Empirical likelihood-based inference under imputation for missing response data. Ann Statist, 2002, 30: 896–924

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang Q H, Rao J N K. Empirical likelihood-based inference in linear model with missing data. Scand J Statist, 2002, 29: 563–576

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang Q H, Wang J L. Inference for the mean difference in the two-sample random censored model. J Multivariate Anal, 2001, 79: 295–315

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang Q H, Zheng Z G. Some asymptotic properties for semiparametric regression models with censored data. Sci China Ser A, 1997, 40: 945–957

    Article  MathSciNet  MATH  Google Scholar 

  31. Xue L G. Empirical likelihood for linear models with missing responses. J Multivariate Anal, 2009, 100: 1353–1366

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang B. Quantile processes in the presence of auxiliary information. Ann Inst Statist Math, 1997, 49: 35–55

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang B. Empirical likelihood confidence intervals for M-functionals in the presence of auxiliary information. Statist Probab Lett, 1997, 32: 87–97

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang J, Cui H J. Empirical likelihood confidence region for parameters in linear EV model with missing data (in Chinese). Acta Math Sci Ser A, 2009, 29: 1465–1476

    MathSciNet  MATH  Google Scholar 

  35. Zhong B, Rao J N K. Empirical likelihood inference under stratified random sampling using auxiliary population information. Biometrika, 2000, 87: 929–938

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhong P, Cui H J. Empirical likelihood for median regression model with designed censoring variables. J Multivariate Anal, 2010, 101: 240–251

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhu L X. Nonparametric Monte Carlo Tests and Their Applications. New York: Springer, 2005

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Financial Mathematics, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    ShanShan Wang

  2. Department of Statistics, School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    HengJian Cui

  3. Department of Statistics, The Pennsylvania State University, University Park, PA, 16802, USA

    RunZe Li

Authors
  1. ShanShan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. HengJian Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. RunZe Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to HengJian Cui.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, S., Cui, H. & Li, R. Empirical likelihood inference for semi-parametric estimating equations. Sci. China Math. 56, 1247–1262 (2013). https://doi.org/10.1007/s11425-012-4494-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-012-4494-8

Keywords

MSC(2010)

Search

Navigation