Bootstrap Consistency for Empirical Likelihood in Density Ratio Models

Weiwei Zhuang, Weiqi Yang and Jiahua Chen

doi:10.5705/ss.202025.0398

Abstract

We establish the validity of bootstrap methods for empirical likelihood (EL) in

ference under the density ratio model (DRM). In particular, we prove that the bootstrap

maximum EL estimators share the same limiting distribution as their population counterparts, both at the parameter level and for distribution functionals. Our results extend

existing pointwise convergence theory to weak convergence of processes, which in turn justifies bootstrap inference for quantiles and dominance indices within the DRM framework.

These theoretical guarantees close an important gap in the literature, providing rigorous

foundations for resampling-based confidence intervals and hypothesis tests. Simulation

studies further demonstrate the accuracy and practical value of the proposed approach.

Key words and phrases: Bootstrap; Density ratio model; Empirical likelihood; Semipara- metric inference; Quantile processes

Information

Preprint No.	SS-2025-0398
Manuscript ID	SS-2025-0398
Complete Authors	Weiwei Zhuang, Weiqi Yang, Jiahua Chen
Corresponding Authors	Jiahua Chen
Emails	jhchen@stat.ubc.ca

References

Anderson, J. (1979). Multivariate logistic compounds. Biometrika 66(1), 17–26.
Bondell, H. D. (2007). Testing goodness-of-fit in logistic case-control studies. Biometrika 94(2), 487–495.
Cai, S., J. Chen, and J. V. Zidek (2017). Hypothesis testing in the presence of multiple samples under density ratio models. Statistica Sinica 27(2), 761–783.
Chen, J. and Y. Liu (2013). Quantile and quantile-function estimations under density ratio model. The Annals of Statistics 41, 1669–1692.
Chen, J. and J. Qin (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika 80, 107–116.
Cheng, G. and J. Z. Huang (2010). Bootstrap consistency for general semiparametric m-estimation. The Annals of Statistics 38(5), 2884.
Chernozhuokov, V., D. Chetverikov, K. Kato, and Y. Koike (2022). Improved central limit theorem and bootstrap approximations in high dimensions. The Annals of Statistics 50(5), 2562–2586.
Davison, A. and D. Hinkley (1997). Bootstrap Methods and Their Application. Cambridge, U.K.: Cambridge University Press.
Efron, B. (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics 7, 1–26.
Efron, B. and R. Tibshirani (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.
Fokianos, K., B. Kedem, J. Qin, and D. A. Short (2001). A semiparametric approach to the one-way layout. Technometrics 43(1), 56–65.
Horowitz, J. L. and A. Krishnamurthy (2018). A bootstrap method for constructing pointwise and uniform confidence bands for conditional quantile funtions. Statistica Sinica 28(4), 2609–2632.
Hu, B. (2018). An R package for monitoring test under density ratio model and its applications. Ph. D. thesis, University of British Columbia.
Huang, A. and P. J. Rathouz (2012). Proportional likelihood ratio models for mean regression. Biometrika 99(1), 223–229.
Kaji, T. (2018). Essays on Asymptotic Methods in Econometrics. Ph. D. thesis, Massachusetts Institute of Technology.
Kedem, B., G. Lu, R. Wei, and P. D. Williams (2008). Forecasting mortality rates via density ratio modeling. Canadian Journal of Statistics 36(2), 193–206.
Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. New York: Springer.
Li, P., Y. Liu, and J. Qin (2017). Semiparametric inference in a genetic mixture model. Journal of the American Statistical Association 112(519), 1250–1260.
Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.
Owen, A. (1991). Empirical likelihood for linear models. The Annals of Statistics 19, 1725–1747.
Parente, P. M. and R. J. Smith (2014). Recent developments in empirical likelihood and related methods. Annu. Rev. Econ. 6(1), 77–102.
Qin, J. (1998). Inferences for case-control and semiparametric two-sample density ratio models. Biometrika 85(3), 619–630.
Qin, J. (2017). Biased Sampling, Over-identified Parameter Problems and Beyond. New York: Springer.
Qin, J. and J. Lawless (1994). Empirical likelihood and general estimating equations. The Annals of Statistics 22(1), 300–325.
Qin, J. and B. Zhang (1997). A goodness-of-fit test for logistic regression models based on case-control data. Biometrika 84(3), 609–618.
Shao, U. and D. Tu (1995). The Jackknife and Bootstrap. New York: Springer-Verlag.
Soukarieh, I. and S. Bouzebda (2023). Renewal type bootstrap for increasing degree u-process of a markov chain. Journal of Multivariate Analysis 195, 105143.
Tan, Z. (2009). A note on profile likelihood for exponential tilt mixture models. Biometrika 96(1), 229–236.
Wu, C. and W. W. Lu (2016). Calibration weighting methods for complex surveys. International Statistical Review 84(1), 79–98.
Wu, C. and R. R. Sitter (2001). A model-calibration approach to using complete auxiliary information from survey data. Journal of the American Statistical Association 96(453), 185–193.
Zhang, A. G. and J. Chen (2022). Density ratio model with data-adaptive basis function. Journal of Multivariate Analysis 191, 105043.
Zhuang, W., B. Hu, and R. Chen (2019). Semiparametric inference for the dominance index under the density ratio model. Biometrika 106(1), 229–241.

Acknowledgments

This work was supported by the National Natural Science Foundation of China

(Grant No. 72571262) and the Natural Sciences and Engineering Research Council

of Canada (RGPIN-2025-03989). The authors also acknowledge the computing

resources provided by the University of British Columbia and the Digital Research

Alliance of Canada.

Supplementary Materials

The online supplementary material contains detailed proofs of the theoretical results (Theorems 1–5 and auxiliary lemmas) established in Sections 3 and 4. It also

provides additional descriptive statistics and empirical distribution plots for the

real-data analysis presented in Section 6.

Supplementary materials are available for download.

[1] Anderson, J. (1979). Multivariate logistic compounds. Biometrika 66(1), 17–26.

[2] Bondell, H. D. (2007). Testing goodness-of-fit in logistic case-control studies. Biometrika 94(2), 487–495.

[3] Cai, S., J. Chen, and J. V. Zidek (2017). Hypothesis testing in the presence of multiple samples under density ratio models. Statistica Sinica 27(2), 761–783.

[4] Chen, J. and Y. Liu (2013). Quantile and quantile-function estimations under density ratio model. The Annals of Statistics 41, 1669–1692.

[5] Chen, J. and J. Qin (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika 80, 107–116.

[6] Cheng, G. and J. Z. Huang (2010). Bootstrap consistency for general semiparametric m-estimation. The Annals of Statistics 38(5), 2884.

[7] Chernozhuokov, V., D. Chetverikov, K. Kato, and Y. Koike (2022). Improved central limit theorem and bootstrap approximations in high dimensions. The Annals of Statistics 50(5), 2562–2586.

[8] Davison, A. and D. Hinkley (1997). Bootstrap Methods and Their Application. Cambridge, U.K.: Cambridge University Press.

[9] Efron, B. (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics 7, 1–26.

[10] Efron, B. and R. Tibshirani (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.

[11] Fokianos, K., B. Kedem, J. Qin, and D. A. Short (2001). A semiparametric approach to the one-way layout. Technometrics 43(1), 56–65.

[12] Horowitz, J. L. and A. Krishnamurthy (2018). A bootstrap method for constructing pointwise and uniform confidence bands for conditional quantile funtions. Statistica Sinica 28(4), 2609–2632.

[13] Hu, B. (2018). An R package for monitoring test under density ratio model and its applications. Ph. D. thesis, University of British Columbia.

[14] Huang, A. and P. J. Rathouz (2012). Proportional likelihood ratio models for mean regression. Biometrika 99(1), 223–229.

[15] Kaji, T. (2018). Essays on Asymptotic Methods in Econometrics. Ph. D. thesis, Massachusetts Institute of Technology.

[16] Kedem, B., G. Lu, R. Wei, and P. D. Williams (2008). Forecasting mortality rates via density ratio modeling. Canadian Journal of Statistics 36(2), 193–206.

[17] Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. New York: Springer.

[18] Li, P., Y. Liu, and J. Qin (2017). Semiparametric inference in a genetic mixture model. Journal of the American Statistical Association 112(519), 1250–1260.

[19] Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.

[20] Owen, A. (1991). Empirical likelihood for linear models. The Annals of Statistics 19, 1725–1747.

[21] Parente, P. M. and R. J. Smith (2014). Recent developments in empirical likelihood and related methods. Annu. Rev. Econ. 6(1), 77–102.

[22] Qin, J. (1998). Inferences for case-control and semiparametric two-sample density ratio models. Biometrika 85(3), 619–630.

[23] Qin, J. (2017). Biased Sampling, Over-identified Parameter Problems and Beyond. New York: Springer.

[24] Qin, J. and J. Lawless (1994). Empirical likelihood and general estimating equations. The Annals of Statistics 22(1), 300–325.

[25] Qin, J. and B. Zhang (1997). A goodness-of-fit test for logistic regression models based on case-control data. Biometrika 84(3), 609–618.

[26] Shao, U. and D. Tu (1995). The Jackknife and Bootstrap. New York: Springer-Verlag.

[27] Soukarieh, I. and S. Bouzebda (2023). Renewal type bootstrap for increasing degree u-process of a markov chain. Journal of Multivariate Analysis 195, 105143.

[28] Tan, Z. (2009). A note on profile likelihood for exponential tilt mixture models. Biometrika 96(1), 229–236.

[29] Wu, C. and W. W. Lu (2016). Calibration weighting methods for complex surveys. International Statistical Review 84(1), 79–98.

[30] Wu, C. and R. R. Sitter (2001). A model-calibration approach to using complete auxiliary information from survey data. Journal of the American Statistical Association 96(453), 185–193.

[31] Zhang, A. G. and J. Chen (2022). Density ratio model with data-adaptive basis function. Journal of Multivariate Analysis 191, 105043.

[32] Zhuang, W., B. Hu, and R. Chen (2019). Semiparametric inference for the dominance index under the density ratio model. Biometrika 106(1), 229–241.