Advertisement

Confidence Intervals and Tests for High-Dimensional Models: A Compact Review

  • Peter BühlmannEmail author
Conference paper
  • 1.8k Downloads
Part of the Lecture Notes in Statistics book series (LNS, volume 217)

Abstract

We present a compact review of methods for constructing tests and confidence intervals in high-dimensional models. Links to theory, finite sample performance results and software allows to obtain a “quick” but sufficiently deep overview for applying the procedures.

Keywords

Multi Sample Sample Splitting Stability Selection Construct Confidence Interval Ridge Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Benjamini, Y., & Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics, 29, 1165–1188.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Benjamini, Y., & Yekutieli, D. (2005). False discovery rate-adjusted multiple confidence intervals for selected parameters. Journal of the American Statistical Association, 100, 71–81.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bühlmann, P. (2013). Statistical significance in high-dimensional linear models. Bernoulli, 19, 1212–1242.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bühlmann, P., & Mandozzi, J. (2013). High-dimensional variable screening and bias in subsequent inference, with an empirical comparison. Computational Statistics. Published online doi: 10.1007/s00180-013-0436-3.
  5. 5.
    Bühlmann, P., Meier, L., & Kalisch, M. (2014). High-dimensional statistics with a view towards applications in biology. Annual Review of Statistics and Its Applications, 1, 255–278.CrossRefGoogle Scholar
  6. 6.
    Bühlmann, P., & van de Geer, S. (2011). Statistics for high-dimensional data: Methods, theory and applications. Heidelberg/New York: Springer.CrossRefGoogle Scholar
  7. 7.
    Chatterjee, A., & Lahiri, S. (2013). Rates of convergence of the adaptive LASSO estimators to the oracle distribution and higher order refinements by the bootstrap. Annals of Statistics, 41, 1232–1259.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dezeure, R., Bühlmann, P., Meier, L., & Meinshausen, N. (2014). High-dimensional inference: Confidence intervals, p-values and software hdi. Preprint arXiv:1408.4026.Google Scholar
  9. 9.
    Diaconis, P., & Freedman, D. (1986). On the consistency of Bayes estimates (with discussion). Annals of Statistics, 14, 1–63.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Javanmard, A., & Montanari, A. (2013). Confidence intervals and hypothesis testing for high-dimensional regression. arXiv:1306.3171.Google Scholar
  11. 11.
    Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Annals of Statistics, 17, 1001–1008.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu, H., & Yu, B. (2013). Asymptotic properties of lasso+mls and lasso+ridge in sparse high-dimensional linear regression. arXiv:1306.5505.Google Scholar
  13. 13.
    Lockhart, R., Taylor, J., Tibshirani, R., & Tibshirani, R. (2014). A significance test for the Lasso. Annals of Statistics, 42(2), 413–468.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Mandozzi, J., & Bühlmann, P. (2013). Hierarchical testing in the high-dimensional setting with correlated variables. arXiv:1312.5556.Google Scholar
  15. 15.
    McCullagh, P., & Nelder, J. (1989). Generalized linear models (2nd ed.). London: Chapman & Hall.zbMATHCrossRefGoogle Scholar
  16. 16.
    Meier, L. (2013). hdi: High-dimensional inference. R package version 0.0-1/r2.Google Scholar
  17. 17.
    Meinshausen, N. (2013). Assumption-free confidence intervals for groups of variables in sparse high-dimensional regression. arXiv:1309.3489.Google Scholar
  18. 18.
    Meinshausen, N., & Bühlmann, P. (2006). High-dimensional graphs and variable selection with the Lasso. Annals of Statistics, 34, 1436–1462.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Meinshausen, N., & Bühlmann, P. (2010). Stability selection (with discussion). Journal of the Royal Statistical Society Series B, 72, 417–473.CrossRefGoogle Scholar
  20. 20.
    Meinshausen, N., Meier, L., & Bühlmann, P. (2009). P-values for high-dimensional regression. Journal of the American Statistical Association, 104, 1671–1681.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Reid, S., Tibshirani, R., & Friedman, J. (2013). A study of error variance estimation in Lasso regression. arXiv:1311.5274.Google Scholar
  22. 22.
    Shao, J., & Deng, X. (2012). Estimation in high-dimensional linear models with deterministic design matrices. Annals of Statistics, 40, 812–831.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Sun, T., & Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika, 99, 879–898.zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    van de Geer, S., Bühlmann, P., Ritov, Y., & Dezeure, R. (2014). On asymptotically optimal confidence regions and tests for high-dimensional models. Annals of Statistics, 42(3), 1166–1202.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Wasserman, L., & Roeder, K. (2009). High dimensional variable selection. Annals of Statistics, 37, 2178–2201.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhang, C.-H., & Zhang, S. (2014). Confidence intervals for low-dimensional parameters in high-dimensional linear models. Journal of the Royal Statistical Society Series B, 76, 217–242.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Seminar for Statistics, ETH ZürichZürichSwitzerland

Personalised recommendations

Cite paper

Buy options