Advertisement

Journal of Systems Science and Complexity

, Volume 30, Issue 5, pp 1206–1226 | Cite as

Generalized F-test for high dimensional regression coefficients of partially linear models

  • Siyang Wang
  • Hengjian CuiEmail author
Article
  • 119 Downloads

Abstract

This paper proposes a test procedure for testing the regression coefficients in high dimensional partially linear models based on the F-statistic. In the partially linear model, the authors first estimate the unknown nonlinear component by some nonparametric methods and then generalize the F-statistic to test the regression coefficients under some regular conditions. During this procedure, the estimation of the nonlinear component brings much challenge to explore the properties of generalized F-test. The authors obtain some asymptotic properties of the generalized F-test in more general cases, including the asymptotic normality and the power of this test with p/n ∈ (0, 1) without normality assumption. The asymptotic result is general and by adding some constraint conditions we can obtain the similar conclusions in high dimensional linear models. Through simulation studies, the authors demonstrate good finite-sample performance of the proposed test in comparison with the theoretical results. The practical utility of our method is illustrated by a real data example.

Keywords

Generalized F-test high dimensional regression partially linear models power of test 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.CrossRefGoogle Scholar
  2. [2]
    Heckman N E, Spline smoothing in a partly linear model, Journal of the Royal Statistical Society: Series B (Methodological), 1986, 48(2): 244–248.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Robinson P M, Root-N-consistent semiparametric regression, Econometrica: Journal of the Econometric Society, 1988, 56(4): 931–954.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Speckman P, Kernel smoothing in partial linear models, Journal of the Royal Statistical Society: Series B (Methodological), 1988, 50(3): 413–436.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Green P J and Silverman B W, Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, CRC Press, Florida, 1993.zbMATHGoogle Scholar
  6. [6]
    Eubank R L, Kambour E L, Kim J T, et al., Estimation in partially linear models, Computational Statistics & Data Analysis, 1998, 29(1): 27–34.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Bunea F and Wegkamp M H, Two-stage model selection procedures in partially linear regression, Canadian Journal of Statistics, 2004, 32(2): 105–118.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Fan J and Li R, New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis, Journal of the American Statistical Association, 2004, 99(467): 710–723.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Xie H and Huang J, SCAD-penalized regression in high-dimensional partially linear models, The Annals of Statistics, 2009, 37(2): 673–696.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Goeman J J, Van De Geer S A, and Van Houwelingen H C, Testing against a high dimensional alternative, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2006, 68(3): 477–493.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Goeman J J, Van Houwelingen H C, and Finos L, Testing against a high-dimensional alternative in the generalized linear model: Asymptotic type I error control, Biometrika, 2011, 98(2): 381–390.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Zhong P S and Chen S, Tests for high-dimensional regression coefficients with factorial designs, Journal of the American Statistical Association, 2011, 106(493): 260–274.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Wang S and Cui H, Generalized F test for high dimensional linear regression coefficients, Journal of Multivariate Analysis, 2013, 117: 134–149.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Lan W, Zhong P S, Li R, et al., Testing a single regression coefficient in high dimensional linear models, http://methodology.psu.edu/media/techreports/13-125.pdf, 2013.zbMATHGoogle Scholar
  15. [15]
    Müller P and Geer S, The partial linear model in high dimensions, Scandinavian Journal of Statistics, 2015, 42(2): 580–608.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Härdle W, Mammen E, and Müller M, Testing parametric versus semiparametric modeling in generalized linear models, Journal of the American Statistical Association, 1998, 93(444): 1461–1474.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Fan J and Huang T, Profile likelihood inferences on semiparametric varying-coefficient partially linear models, Bernoulli, 2005, 11(6): 1031–1057.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Huang L S and Davidson P W, Analysis of variance and F-tests for partial linear models with applications to environmental health data, Journal of the American Statistical Association, 2010, 105(491): 991–1004.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Härdie W, Liang H, and Gao J, Partially Linear Models, Physica-Verlag, Heidelberg, 2000.CrossRefGoogle Scholar
  20. [20]
    Gao J, Nonlinear Time Series: Semiparametric and Nonparametric Methods, CRC Press, Florida, 2007.CrossRefzbMATHGoogle Scholar
  21. [21]
    Zhang C H and Huang J, The sparsity and bias of the lasso selection in high-dimensional linear regression, The Annals of Statistics, 2008, 36(4): 1567–1594.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Chiang A P, Beck J S, Yen H J, et al., Homozygosity mapping with SNP arrays identifies TRIM32, an E3 ubiquitin ligase, as a BardetCBiedl syndrome gene (BBS11), Proceedings of the National Academy of Sciences, 2006, 103(16): 6287–6292.CrossRefGoogle Scholar
  23. [23]
    Scheetz T E, Kim K Y A, Swiderski R E, et al., Regulation of gene expression in the mammalian eye and its relevance to eye disease, Proceedings of the National Academy of Sciences, 2006, 103(39): 14429–14434.CrossRefGoogle Scholar
  24. [24]
    Huang J, Ma S, and Zhang C H, Adaptive Lasso for sparse high-dimensional regression models, Statistica Sinica, 2008, 18(4): 1603–1618.MathSciNetzbMATHGoogle Scholar
  25. [25]
    Bhansali R J, Giraitis L, and Kokoszka P S, Convergence of quadratic forms with nonvanishing diagonal, Statistics & Probability Letters, 2007, 77(7): 726–734.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Zhu L X, Nonparametric Monte Carlo Tests and Their Applications, Springer, New York, 2005.zbMATHGoogle Scholar
  27. [27]
    Zou H and Zhang H H, On the adaptive elastic-net with a diverging number of parameters, Annals of Statistics, 2009, 37(4): 1733.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Li G, Peng H, and Zhu L, Nonconcave penalized M-estimation with a diverging number of parameters, Statistica Sinica, 2011, 21(1): 391–419.MathSciNetzbMATHGoogle Scholar
  29. [29]
    Bai Z and Saranadasa H, Effect of high dimension: By an example of a two sample problem, Statistica Sinica, 1996, 6: 311–329.MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Statistics and MathematicsCentral University of Finance and EconomicsBeijingChina
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijingChina

Personalised recommendations