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Empirical Likelihood Test for Regression Coefficients in High Dimensional Partially Linear Models

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Abstract

This paper considers tests for regression coefficients in high dimensional partially linear Models. The authors first use the B-spline method to estimate the unknown smooth function so that it could be linearly expressed. Then, the authors propose an empirical likelihood method to test regression coefficients. The authors derive the asymptotic chi-squared distribution with two degrees of freedom of the proposed test statistics under the null hypothesis. In addition, the method is extended to test with nuisance parameters. Simulations show that the proposed method have a good performance in control of type-I error rate and power. The proposed method is also employed to analyze a data of Skin Cutaneous Melanoma (SKCM).

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References

  1. Engle R F, Granger C W J, Rice J, et al., Semiparametric estimates of the relation between weather and electricity sales, Journal of the American Statistical Association, 1986, 81(394): 310–320.

    Article  Google Scholar 

  2. Zhong P S and Chen S X, Tests for high-dimensional regression coefficients with factorial designs, Journal of the American Statistical Association, 2011, 106(493): 260–274.

    Article  MathSciNet  Google Scholar 

  3. Owen A B, Empirical likelihood ratio confidence intervals for a single functional, Biometrika, 1988, 75(2): 237–249.

    Article  MathSciNet  Google Scholar 

  4. Peng L, Qi Y C, and Wang R D, Empirical likelihood test for high dimensional linear models, Statistics and Probability Letters, 2014, 86: 85–90.

    Article  MathSciNet  Google Scholar 

  5. Zang Y G, Zhang Q Z, Zhang S G, et al., Empirical likelihood test for high dimensional generalized linear models, Big and Complex Data Analysis, Springer and Cham, 2017.

    Google Scholar 

  6. Wang R D, Peng L, and Qi Y C, Jackknife empirical likelihood test for equality of two high dimensional means, Statistica Sinica, 2013, 23(2): 667–690.

    MathSciNet  MATH  Google Scholar 

  7. Zang Y G, Zhang S G, Li Q Z, et al., Jackknife empirical likelihood test for high-dimensional regression coefficients, Computational Statistics and Data Analysis, 2016, 94: 302–316.

    Article  MathSciNet  Google Scholar 

  8. Wang Q H and Rao J N K, Empirical likelihood-based inference under imputation for missing response data, Annals of Statistics, 2002, 30(3): 896–924.

    Article  MathSciNet  Google Scholar 

  9. Xue L G and Zhu L, Empirical likelihood for single-index models, Journal of Multivariate Analysis, 2006, 97(6): 1295–1312.

    Article  MathSciNet  Google Scholar 

  10. Agarwal G G and Studden W J, Asymptotic integrated mean square error using least squares and bias minimizing splines, The Annals of Statistics, 1980, 8(6): 1307–1325.

    Article  MathSciNet  Google Scholar 

  11. Stone C J, Optimal rates of convergence for nonparametric estimators, The Annals of Statistics, 1980, 8(6): 1348–1360.

    Article  MathSciNet  Google Scholar 

  12. Stone C J, Optimal global rates of convergence for nonparametric regression, The Annals of Statistics, 1982, 10(4): 1040–1053.

    Article  MathSciNet  Google Scholar 

  13. Newey W K, Convergence rates and asymptotic normality for series estimators, Journal of Econometrics, 1997, 79(1): 147–168.

    Article  MathSciNet  Google Scholar 

  14. Linton O, Second order approximation in the partially linear regression model, Econometrica, 1995, 63(6): 1079–1112.

    Article  MathSciNet  Google Scholar 

  15. Shi P D and Li G Y, A note on the convergent rates of M-estimates for a partly linear model, Statistics, 1995, 26(1): 27–47.

    Article  MathSciNet  Google Scholar 

  16. Donald S G and Newey W K, Series estimation of semilinear models, Journal of Multivariate Analysis, 1994, 50(1): 30–40.

    Article  MathSciNet  Google Scholar 

  17. Wang S Y and Cui H J, Generalized F-test for high dimensional regression coefficients of partially linear models, Journal of Systems Science and Complexity, 2017, 30(5): 1206–1226.

    Article  MathSciNet  Google Scholar 

  18. Bai Z D and Saranadasa H, Effect of high dimension: By an example of a two sample problem, Statistica Sinica, 1996, 6(2): 311–329.

    MathSciNet  MATH  Google Scholar 

  19. Owen A B, Empirical Likelihood, Chapman and Hall, 2001.

  20. Bravo F, Second-order power comparisons for a class of nonparametric likelihood based tests, Biometrika, 2003, 90(4): 881–890.

    Article  MathSciNet  Google Scholar 

  21. Stark A, Stahl M S, Kirchner H L, et al., Body mass index at the time of diagnosis and the risk of advanced stages and poorly differentiated cancers of the breast: Findings from a case-series study, International Journal of Obesity, 2010, 34(9): 1381–1386.

    Article  Google Scholar 

  22. Fan J Q and Song R, Sure independence screening in generalized linear models with NP-dimensionality, The Annals of Statistics, 2010, 38(6): 3567–3604.

    Article  MathSciNet  Google Scholar 

  23. Huang J, Ma S G, and Zhang C H, The Iterated lasso for high-dimensional logistic regression, The University of Iowa Department of Statistical and Actuarial Science Technical Report, 2008.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yan Liu, Mingyang Ren & Sanguo Zhang

  2. Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing, 100190, China

    Yan Liu, Mingyang Ren & Sanguo Zhang

Authors
  1. Yan Liu
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  2. Mingyang Ren
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  3. Sanguo Zhang
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Corresponding author

Correspondence to Sanguo Zhang.

Additional information

This work is supported by the University of Chinese Academy of Sciences under Grant No. Y95401TXX2, Beijing Natural Science Foundation under Grant No. Z190004, and Key Program of Joint Funds of the National Natural Science Foundation of China under Grant No. U19B2040.

This paper was recommended for publication by Editor DONG Yuexiao.

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Liu, Y., Ren, M. & Zhang, S. Empirical Likelihood Test for Regression Coefficients in High Dimensional Partially Linear Models. J Syst Sci Complex 34, 1135–1155 (2021). https://doi.org/10.1007/s11424-020-9260-3

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  • DOI: https://doi.org/10.1007/s11424-020-9260-3

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