Electronic Journal of Statistics

Maximum empirical likelihood estimation and related topics

Hanxiang Peng and Anton Schick

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This article develops a theory of maximum empirical likelihood estimation and empirical likelihood ratio testing with irregular and estimated constraint functions that parallels the theory for parametric models and is tailored for semiparametric models. The key is a uniform local asymptotic normality condition for the local empirical likelihood ratio. This condition is shown to hold under mild assumptions on the constraint function. Applications of our results are discussed to inference problems about quantiles under possibly additional information on the underlying distribution and to residual-based inference about quantiles.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 2962-2994.

Received: February 2017
First available in Project Euclid: 19 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties

Irregular and estimated constraints nuisance parameters empirical likelihood ratio tests uniform local asymptotic normality condition

Creative Commons Attribution 4.0 International License.


Peng, Hanxiang; Schick, Anton. Maximum empirical likelihood estimation and related topics. Electron. J. Statist. 12 (2018), no. 2, 2962--2994. doi:10.1214/18-EJS1471. https://projecteuclid.org/euclid.ejs/1537344589

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  • [1] Fabian, V. and Hannan, J. (1982). On estimation and adaptive estimation for locally asymptotically normal families., Z. Wahrsch. Verw. Gebiete 59 459–478.
  • [2] Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. With a discussion by J. A. Wellner and J. Praestgaard and a reply by the author., Scand. J. Statist. 16 97–128.
  • [3] Hjort, N. L., McKeague, I. W. and Van Keilegom, I. (2009). Extending the scope of empirical likelihood., Ann. Statist. 37 1079–1111.
  • [4] Koul, H. L. (1969). Asymptotic behavior of Wilcoxon type confidence regions in multiple linear regression., Ann. Statist. 40 1950–1979.
  • [5] Koul, H. L., Müller, U.U. and Schick, A. (2017). Estimating the error distribution in a single-index model., From Statistics to Mathematical Finance, Festschrift in Honour of Winfried Stute (D. Ferger, W. González Manteiga, T. Schmidt, J.-L. Wang, eds.), 209–233, Springer, Cham.
  • [6] Le Cam, L. (1960). Locally asymptotically normal families of distributions., Univ. California Publ. Statist. 3 37–98.
  • [7] Molanes Lopez, E. M., Van Keilegom, I. and Veraverbeke, N. (2009). Empirical likelihood for non-smooth criterion functions., Scand. J. Statist. 36 413–432.
  • [8] Müller, U. U., Schick A. and Wefelmeyer, W. (2007). Estimating the error distribution function in semiparametric regression., Statist. Decisions 25 1–18.
  • [9] Müller, U. U., Schick A. and Wefelmeyer, W. (2009). Estimating the error distribution function in nonparametric regression with multivariate covariates., Statist. Probab. Lett. 79 957–964.
  • [10] Müller, U. U., Schick A. and Wefelmeyer, W. (2012). Estimating the error distribution function in semiparametric additive regression models., J. Statist. Plann. Inference 142 552–566.
  • [11] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional., Biometrika 75 237–249.
  • [12] Owen, A. B. (1990). Empirical likelihood ratio confidence regions., Ann. Sta-tist. 18 90–120.
  • [13] Owen, A. B. (2001)., Empirical Likelihood. Chapman & Hall/CRC, London.
  • [14] Parente, P. M. D. C. and Smith, R. J. (2011). GEL methods for nonsmooth moment indicators., Econometric Theory 27 74–113.
  • [15] Peng, H. and Schick, A. (2013). Empirical likelihood approach to goodness of fit testing., Bernoulli 19 954–981.
  • [16] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations., Ann. Statist. 22 300–325.
  • [17] R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria., http://www.R-project.org/.
  • [18] Xue, L. and Wang, Q. (2012). Empirical likelihood for single-index varying-coefficient models., Bernoulli 18 836–856.
  • [19] Xue, L. and Zhu, L. (2006) Empirical likelihood for single-index models., J. Multivariate Anal. 97 1295–1312.
  • [20] Zhu, L. and Xue, L. (2006). Empirical likelihood confidence regions in a partially linear single-index model., J. Roy. Statist. Soc., Series B 68 549–570.