The Annals of Statistics

Empirical likelihood-based inference under imputation for missing response data

J. N. K. Rao and Qihua Wang

Full-text: Open access

Abstract

Inference under kernel regression imputation for missing response data is considered. An adjusted empirical likelihood approach to inference for the mean of the response variable is developed. A nonparametric version of Wilks' theorem is proved for the adjusted empirical log-likelihood ratio by showing that it has an asymptotic standard chi-squared distribution, and the corresponding empirical likelihood confidence interval for the mean is constructed. With auxiliary information, an empirical likelihood-based estimator is defined and an adjusted empirical log-likelihood ratio is derived. Asymptotic normality of the estimator is proved. Also, it is shown that the adjusted empirical log-likelihood ratio obeys Wilks' theorem. A simulation study is conducted to compare the adjusted empirical likelihood and the normal approximation methods in terms of coverage accuracies and average lengths of confidence intervals. Based on biases and standard errors, a comparison is also made by simulation between the empirical likelihood-based estimator and related estimators. Our simulation indicates that the adjusted empirical likelihood method performs competitively and that the use of auxiliary information provides improved inferences.

Article information

Source
Ann. Statist., Volume 30, Number 3 (2002), 896-924.

Dates
First available in Project Euclid: 6 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1028674845

Digital Object Identifier
doi:10.1214/aos/1028674845

Mathematical Reviews number (MathSciNet)
MR1922545

Zentralblatt MATH identifier
1029.62040

Subjects
Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory

Keywords
Empirical likelihood missing response regression imputation

Citation

Wang, Qihua; Rao, J. N. K. Empirical likelihood-based inference under imputation for missing response data. Ann. Statist. 30 (2002), no. 3, 896--924. doi:10.1214/aos/1028674845. https://projecteuclid.org/euclid.aos/1028674845


Export citation

References

  • ADIMARI, G. (1997). Empirical likelihood ty pe confidence intervals under random censorship. Ann. Inst. Statist. Math. 49 447-466.
  • CHEN, J. H. and QIN, J. (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika 80 107-116.
  • CHEN, J. H. and SHAO, J. (2000). Nearest neighbor imputation for survey data. J. Official Statist. 16 113-131.
  • CHEN, S. X. (1993). On the accuracy of empirical likelihood confidence regions for linear regression model. Ann. Inst. Statist. Math. 45 621-637.
  • CHEN, S. X. (1994). Empirical likelihood confidence intervals for linear regression coefficients. J. Multivariate Anal. 49 24-40.
  • CHEN, S. X. and HALL, P. (1993). Smoothed empirical likelihood confidence intervals for quantiles. Ann. Statist. 21 1166-1181.
  • CHENG, P. E. (1994). Nonparametric estimation of mean functionals with data missing at random. J. Amer. Statist. Assoc. 89 81-87.
  • DICICCIO, T. J., HALL, P. and ROMANO, J. P. (1991). Empirical likelihood is Bartlett-correctable. Ann. Statist. 19 1053-1061.
  • HALL, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • HALL, P. and LA SCALA, B. (1990). Methodology and algorithms of empirical likelihood. Internat. Statist. Rev. 58 109-127.
  • HARTLEY, H. O. and RAO, J. N. K. (1968). A new estimation theory for sample survey s. Biometrika 55 547-557.
  • HEALY, M. J. R. and WESTMACOTT, M. (1956). Missing values in experiments analysed on automatic computers. Appl. Statist. 5 203-206.
  • KITAMURA, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084-2102.
  • KOLACZy K, E. D. (1994). Empirical likelihood for generalized linear models. Statist. Sinica 4 199- 218.
  • KONG, A., LIU, J. S. and WONG, W. H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 89 278-288.
  • LITTLE, R. J. A. and RUBIN, D. B. (1987). Statistical Analy sis with Missing Data. Wiley, New York.
  • OWEN, A. (1988). Empirical likelihood ratio confidence intervals for single functional. Biometrika 75 237-249.
  • OWEN, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90-120.
  • OWEN, A. (1991). Empirical likelihood for linear models. Ann. Statist. 19 1725-1747.
  • QIN, J. (1993). Empirical likelihood in biased sample problems. Ann. Statist. 21 1182-1196.
  • QIN, J. and LAWLESS, J. F. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300-325.
  • RAO, J. N. K. (1996). On variance estimation with imputed survey data (with discussion). J. Amer. Statist. Assoc. 91 499-520.
  • RAO, J. N. K. and SHAO, J. (1992). Jacknife variance estimation with survey data under hot deck imputation. Biometrika 79 811-822.
  • THOMAS, D. R. and GRUNKEMEIER, G. L. (1975). Confidence interval estimation of survival probabilities for censored data. J. Amer. Statist. Assoc. 70 865-871.
  • WANG, Q. H. and JING, B. Y. (1999). Empirical likelihood for partial linear models with fixed designs. Statist. Probab. Lett. 41 425-433.
  • YATES, F. (1933). The analysis of replicated experiments when the field results are incomplete. Empire Journal of Experimental Agriculture 1 129-142.
  • ZHANG, B. (1997). Empirical likelihood confidence intervals for M-functionals in the presence of auxiliary information. Statist. Probab. Lett. 32 87-97.