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Empirical Likelihood for Non‐Smooth Criterion Functions

Author

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  • ELISA M. MOLANES LOPEZ
  • INGRID VAN KEILEGOM
  • NOËL VERAVERBEKE

Abstract

. Suppose that X1,…, Xn is a sequence of independent random vectors, identically distributed as a d‐dimensional random vector X. Let be a parameter of interest and be some nuisance parameter. The unknown, true parameters (μ0,ν0) are uniquely determined by the system of equations E{g(X,μ0,ν0)} = 0, where g = (g1,…,gp+q) is a vector of p+q functions. In this paper we develop an empirical likelihood (EL) method to do inference for the parameter μ0. The results in this paper are valid under very mild conditions on the vector of criterion functions g. In particular, we do not require that g1,…,gp+q are smooth in μ or ν. This offers the advantage that the criterion function may involve indicators, which are encountered when considering, e.g. differences of quantiles, copulas, ROC curves, to mention just a few examples. We prove the asymptotic limit of the empirical log‐likelihood ratio, and carry out a small simulation study to test the performance of the proposed EL method for small samples.

Suggested Citation

  • Elisa M. Molanes Lopez & Ingrid Van Keilegom & Noël Veraverbeke, 2009. "Empirical Likelihood for Non‐Smooth Criterion Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 413-432, September.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:3:p:413-432
    DOI: 10.1111/j.1467-9469.2009.00640.x
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    References listed on IDEAS

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    1. Chen, Jian & Peng, Liang & Zhao, Yichuan, 2009. "Empirical likelihood based confidence intervals for copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 137-151, January.
    2. Sherman, Robert P, 1993. "The Limiting Distribution of the Maximum Rank Correlation Estimator," Econometrica, Econometric Society, vol. 61(1), pages 123-137, January.
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    Cited by:

    1. Gong, Yun & Peng, Liang & Qi, Yongcheng, 2010. "Smoothed jackknife empirical likelihood method for ROC curve," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1520-1531, July.
    2. Liang Peng & Yongcheng Qi & Ingrid Van Keilegom, 2012. "Jackknife empirical likelihood method for copulas," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(1), pages 74-92, March.
    3. Liang Peng & Yongcheng Qi, 2010. "Smoothed jackknife empirical likelihood method for tail copulas," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 514-536, November.
    4. Yongli Sang & Xin Dang & Yichuan Zhao, 2020. "Depth-based weighted jackknife empirical likelihood for non-smooth U-structure equations," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(2), pages 573-598, June.
    5. Xu, Ke-Li, 2020. "Inference of local regression in the presence of nuisance parameters," Journal of Econometrics, Elsevier, vol. 218(2), pages 532-560.
    6. Song Chen & Ingrid Van Keilegom, 2009. "A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 415-447, November.
    7. Letón, Emilio & Molanes, Elisa M., 2009. "Adjusted empirical likelihood estimation of the youden index and associated threshold for the bigamma model," DES - Working Papers. Statistics and Econometrics. WS ws091907, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. Jing Sun, 2020. "An improvement on the efficiency of complete-case-analysis with nonignorable missing covariate data," Computational Statistics, Springer, vol. 35(4), pages 1621-1636, December.

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