Skip to main content
SpringerLink
Log in

A review on empirical likelihood methods for regression

  • Invited Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

We provide a review on the empirical likelihood method for regression-type inference problems. The regression models considered in this review include parametric, semiparametric, and nonparametric models. Both missing data and censored data are accommodated.

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

  • Barndorff-Nielsen OE, Cox DR (1984) Bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator. J R Stat Soc Ser B 46:483–495

    MATH  MathSciNet  Google Scholar 

  • Barndorff-Nielsen OE, Hall PG (1988) On the level-error after Bartlett adjustment of the likelihood ratio statistic. Biometrika 75:374–378

    Article  MATH  MathSciNet  Google Scholar 

  • Bartlett MS (1937) Properties of sufficiency and statistical tests. Proc R Soc A 160:268–282

    Article  Google Scholar 

  • Buckley JJ, James IR (1979) Linear regression with censored data. Biometrika 66:429–436

    Article  MATH  Google Scholar 

  • Chen SX (1993) On the accuracy of empirical likelihood confidence regions for linear regression model. Ann Inst Stat Math 45:621–637

    Article  MATH  Google Scholar 

  • Chen SX (1994) Empirical likelihood confidence intervals for linear regression coefficients. J Multivar Anal 49:24–40

    Article  MATH  Google Scholar 

  • Chen SX (1996) Empirical likelihood confidence intervals for nonparametric density estimation. Biometrika 83:329–341

    Article  MATH  MathSciNet  Google Scholar 

  • Chen SX, Cui H (2003) An extended empirical likelihood for generalized linear models. Stat Sin 13:69–81

    MATH  MathSciNet  Google Scholar 

  • Chen SX, Cui H-J (2006) On Bartlett correction of empirical likelihood in the presence of nuisance parameters. Biometrika 93:215–220

    Article  MATH  MathSciNet  Google Scholar 

  • Chen SX, Cui H-J (2007) On the second order properties of empirical likelihood with moment restrictions. J Econom 141:492–516

    Article  MathSciNet  Google Scholar 

  • Chen SX, Qin Y-S (2000) Empirical likelihood confidence interval for a local linear smoother. Biometrika 87:946–953

    Article  MATH  MathSciNet  Google Scholar 

  • Chen SX, Qin Y-S (2003) Coverage accuracy of confidence intervals in nonparametric regression. Acta Math Appl Sin Engl Ser 19:387–396

    Article  MATH  MathSciNet  Google Scholar 

  • Chen SX, Van Keilegom I (2009) A goodness-of-fit test for parametric and semiparametric models in multiresponse regression. Bernoulli (to appear)

  • Chen SX, Härdle W, Li M (2003a) An empirical likelihood goodness-of-fit test for time series. J R Stat Soc, Ser B 65:663–678

    Article  MATH  Google Scholar 

  • Chen SX, Leung DHY, Qin J (2003b) Information recovery in a study with surrogate endpoints. J Am Stat Assoc 98:1052–1062

    Article  MATH  MathSciNet  Google Scholar 

  • Chen SX, Leung DHY, Qin J (2008) Improved semiparametric estimation using surrogate data. J R Stat Soc, Ser B 70:803–823

    Article  MATH  MathSciNet  Google Scholar 

  • DiCiccio T, Hall P, Romano J (1989) Comparison of parametric and empirical likelihood functions. Biometrika 76:465–476

    Article  MATH  MathSciNet  Google Scholar 

  • DiCiccio T, Hall P, Romano J (1991) Empirical likelihood is Bartlett-correctable. Ann Stat 19:1053–1061

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman & Hall, London

    MATH  Google Scholar 

  • Fan J, Zhang J (2004) Sieve empirical likelihood ratio tests for nonparametric functions. Ann Stat 32:1858–1907

    MathSciNet  Google Scholar 

  • Fan J, Zhang C, Zhang J (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Ann Stat 29:153–193

    Article  MATH  MathSciNet  Google Scholar 

  • Hall P (1991) Edgeworth expansions for nonparametric density estimators, with applications. Statistics 22:215–232

    Article  MATH  MathSciNet  Google Scholar 

  • Hall P, La Scala B (1990) Methodology and algorithms of empirical likelihood. Int Stat Rev 58:109–127

    Article  MATH  Google Scholar 

  • Hansen LP (1982) Large sample properties of generalized method of moments estimators. Econometrica 50:1029–1054

    Article  MATH  MathSciNet  Google Scholar 

  • Härdle W (1990) Applied Nonparametric Regression. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947

    Google Scholar 

  • Hartley HO, Rao JNK (1968) A new estimation theory for sample surveys. Biometrika 55:547–557

    Article  MATH  Google Scholar 

  • Hayakawa T (1977) The likelihood ratio criterion and the asymptotic expansion of its distribution. Ann Inst Stat Math 29:359–378

    Article  MATH  MathSciNet  Google Scholar 

  • Heuchenne C, Van Keilegom I (2007) Polynomial regression with censored data based on preliminary nonparametric estimation. Ann Inst Stat Math 59:273–298

    Article  MATH  Google Scholar 

  • Hjort NL, McKeague IW, Van Keilegom I (2009) Extending the scope of empirical likelihood. Ann Stat 37:1079–1115

    Article  MATH  Google Scholar 

  • Hu X, Wang Z, Zhao Z (2009) Empirical likelihood for semiparametric varying-coefficient partially linear errors-in-variables models. Stat Prob Lett 79:1044–1052

    Article  MATH  MathSciNet  Google Scholar 

  • Kolaczyk ED (1994) Empirical likelihood for generalized linear model. Stat Sin 4:199–218

    MATH  MathSciNet  Google Scholar 

  • Koul H, Susarla V, Van Ryzin J (1981) Regression analysis with randomly right-censored data. Ann Stat 9:1276–1288

    Article  MATH  Google Scholar 

  • Lawley DN (1956) A general method for approximating the distribution of likelihood ratio criteria. Biometrika 43:295–303

    MATH  MathSciNet  Google Scholar 

  • Li G, Van Keilegom I (2002) Likelihood ratio confidence bands in nonparametric regression with censored data. Scand J Stat 29:547–562

    Article  MATH  Google Scholar 

  • Li G, Wang Q-H (2003) Empirical likelihood regression analysis for right censored data. Stat Sin 13:51–68

    MATH  Google Scholar 

  • Liang H, Qin Y (2008) Empirical likelihood-based inferences for partially linear models with missing covariates. Aust N Z J Stat 50:347–359

    Article  MathSciNet  Google Scholar 

  • Lu W, Liang Y (2006) Empirical likelihood inference for linear transformation models. J Multivar Anal 97:1586–1599

    Article  MATH  MathSciNet  Google Scholar 

  • Lu X (2009) Empirical likelihood for heteroscedastic partially linear models. J Multivar Anal 100:387–396

    Article  MATH  Google Scholar 

  • McCullagh P, Nelder JA (1983) Generalized linear models. Chapman & Hall, London

    MATH  Google Scholar 

  • Molanes-López E, Van Keilegom I, Veraverbeke N (2009) Empirical likelihood for non-smooth criterion functions. Scand J Stat 36:413–432

    Article  MathSciNet  Google Scholar 

  • Owen AB (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75:237–249

    Article  MATH  MathSciNet  Google Scholar 

  • Owen AB (1990) Empirical likelihood confidence regions. Ann Stat 18:90–120

    Article  MATH  MathSciNet  Google Scholar 

  • Owen AB (1991) Empirical likelihood for linear models. Ann Stat 19:1725–1747

    Article  MATH  MathSciNet  Google Scholar 

  • Owen A (2001) Empirical likelihood. Chapman & Hall, New York

    MATH  Google Scholar 

  • Tripathi G, Kitamura Y (2003) Testing conditional moment restrictions. Ann Stat 31:2059–2095

    Article  MATH  MathSciNet  Google Scholar 

  • Qin G, Jing B-Y (2001a) Empirical likelihood for censored linear regression. Scand J Stat 28:661–673

    Article  MATH  MathSciNet  Google Scholar 

  • Qin G, Jing B-Y (2001b) Censored partial linear models and empirical likelihood. J Multivar Anal 78:37–61

    Article  MATH  MathSciNet  Google Scholar 

  • Qin G, Jing B-Y (2001c) Empirical likelihood for Cox regression model under random censoring. Commun Stat Simul Comput 30:79–90

    Article  MATH  MathSciNet  Google Scholar 

  • Qin G, Tsao M (2003) Empirical likelihood inference for median regression models for censored survival data. J Multivar Anal 85:416–430

    Article  MATH  MathSciNet  Google Scholar 

  • Qin J, Lawless J (1994) Empirical likelihood and general estimating equations. Ann Stat 22:300–325

    Article  MATH  MathSciNet  Google Scholar 

  • Qin J, Zhang B (2007) Empirical-likelihood-based inference in missing response problems and its application in observational studies. J R Stat Soc, Ser B 69:101–122

    MathSciNet  Google Scholar 

  • Rosenbaum PR, Rubin DB (1983) The central role of the propensity score in observational studies for causal effects. Biometrika 70:41–55

    Article  MATH  MathSciNet  Google Scholar 

  • Rubin DB (1976) Inference and missing data. With comments by RJA Little and a reply by the author. Biometrika 63:581–592

    Article  MATH  MathSciNet  Google Scholar 

  • Shi J, Lau T-S (2000) Empirical likelihood for partially linear models. J Multivar Anal 72:132–148

    Article  MATH  MathSciNet  Google Scholar 

  • Thomas DR, Grunkemeier GL (1975) Confidence interval estimation of survival probabilities for censored data. J Am Stat Assoc 70:865–871

    Article  MATH  MathSciNet  Google Scholar 

  • Tsao M (2004) Bounds on coverage probabilities of the empirical likelihood ratio confidence regions. Ann Stat 32:1215–1221

    Article  MATH  MathSciNet  Google Scholar 

  • Van Keilegom I, Sánchez Sellero C, González Manteiga W (2008) Empirical likelihood based testing for regression. Electr J Stat 2:581–604

    Article  Google Scholar 

  • Wang D, Chen SX (2009) Empirical likelihood for estimating equation with missing values. Ann Stat 37:490–517

    Article  MATH  Google Scholar 

  • Wang L, Veraverbeke N (2006) Empirical likelihood in a semi-parametric model for missing response data. Commun Stat Theory Methods 35:625–639

    Article  MATH  MathSciNet  Google Scholar 

  • Wang Q-H, Jing B-Y (2003) Empirical likelihood for partial linear models. Ann Inst Stat Meth 55:585–595

    Article  MATH  MathSciNet  Google Scholar 

  • Wang Q-H, Li G (2002) Empirical likelihood semiparametric regression analysis under random censorship. J Multivar Anal 83:469–486

    Article  MATH  Google Scholar 

  • Wang Q, Rao JNK (2002) Empirical likelihood-based inference under imputation for missing response data. Dedicated to the memory of Lucien Le Cam. Ann Stat 30:896–924

    Article  MATH  MathSciNet  Google Scholar 

  • Wang Q, Linton O, Härdle W (2004) Semiparametric regression analysis with missing response at random. J Am Stat Assoc 99:334–345

    Article  MATH  Google Scholar 

  • Whang Y-J (2006) Smoothed empirical likelihood methods for quantile regression models. Econom Theory 22:173–205

    Article  MATH  MathSciNet  Google Scholar 

  • Wilks SS (1938) The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann Math Stat 9:60–62

    Article  MATH  Google Scholar 

  • Xue L-G, Zhu L (2006) Empirical likelihood for single-index models. J Multivar Anal 97:1295–1312

    Article  MATH  MathSciNet  Google Scholar 

  • Ying Z, Jung SH, Wei LJ (1995) Survival analysis with median regression models. J Am Stat Assoc 90:178–184

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao Y, Chen F (2008) Empirical likelihood inference for censored median regression model via nonparametric kernel estimation. J Multivar Anal 99:215–231

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao Y, Wang H (2008) Empirical likelihood inference for the regression model of mean quality-adjusted lifetime with censored data. Can J Stat 36:463–478

    Article  MATH  Google Scholar 

  • Zhou M (1992) M-estimation in censored linear models. Biometrika 79:837–841

    MATH  MathSciNet  Google Scholar 

  • Zhou M (2005) Empirical likelihood analysis of the rank estimator for the censored accelerated failure time model. Biometrika 92:492–498

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou M, Li G (2008) Empirical likelihood analysis of the Buckley–James estimator. J Multivar Anal 99:649–664

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou XH, Qin GS, Lin HZ, Li G (2006) Inferences in censored cost regression models with empirical likelihood. Stat Sin 16:1213–1232

    MATH  MathSciNet  Google Scholar 

  • Zhu L, Xue L (2006) Empirical likelihood confidence regions in a partially linear single-index model. J R Stat Soc, Ser B 68:549–570

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Iowa State University, Ames, IA, 50011-1210, USA

    Song Xi Chen

  2. Guanghua School of Management, Peking University, Beijing, China

    Song Xi Chen

  3. Institute of Statistics, Université catholique de Louvain, Voie du Roman Pays 20, 1348, Louvain-la-Neuve, Belgium

    Ingrid Van Keilegom

Authors
  1. Song Xi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ingrid Van Keilegom
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ingrid Van Keilegom.

Additional information

This invited paper is discussed in the comments available at: doi:10.1007/s11749-009-0160-z, doi:10.1007/s11749-009-0161-y, doi:10.1007/s11749-009-0162-x, doi:10.1007/s11749-009-0163-9, doi:10.1007/s11749-009-0164-8, doi:10.1007/s11749-009-0165-7.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, S.X., Van Keilegom, I. A review on empirical likelihood methods for regression. TEST 18, 415–447 (2009). https://doi.org/10.1007/s11749-009-0159-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-009-0159-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation