Skip to main content
SpringerLink
Log in

Non-parametric confidence intervals for covariance and correlation

  • Published:
METRON Aims and scope Submit manuscript

Abstract

Consider a sample of independent and identical bivariate observations. Simple consistent confidence intervals for the variances, covariance, and correlation of the underlying population are obtained from their influence functions. They contrast with their confidence intervals obtained under the assumption of normality, which are shown to be not consistent if the assumption of normality is false. Even when the marginals are normal, we show that Fisher’s \(z\)-transformation may be quite inappropriate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H., Tukey, J.W.: Robust Estimates of Location: Survey and Advances. Princeton University Press, Princeton (1972)

    MATH  Google Scholar 

  2. Azzalini, A., Dalla Valle, A.: The multivariate skew-normal distribution. Biometrika 83, 715–726 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chung, K.L.: A Course in Probability Theory, 2nd edn. Academic Press, New York (1974)

    MATH  Google Scholar 

  4. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 2. Wiley, New York (1962)

    MATH  Google Scholar 

  5. Dette, H., Neumeyer, N.: Nonparametric analysis of covariance. Ann. Stat. 29, 1361–1400 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, Y., Akritas, M.G., Van Keilegom, I.: Nonparametric analysis of covariance for censored data. Biometrika 90, 269–287 (2003)

    Google Scholar 

  7. Efron, B.: Better bootstrap confidence intervals. J. Am. Stat. Assoc. 82, 171–185 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Efron, B.: Bootstrap confidence intervals: good or bad? Psychol. Bull. 104, 293–296 (1988)

    Article  Google Scholar 

  9. Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap. Taylor and Francis, New York (1994)

    Google Scholar 

  10. Guo, Y., Manatunga, A.K.: Nonparametric estimation of the concordance correlation coefficient under univariate censoring. Biometrics 63, 164–172 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hall, P., Martin, M.A., Schucany, W.R.: Better nonparametric bootstrap confidence intervals for the correlation coefficient. J. Stat. Comput. Simul. 33, 161–172 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huber, P.J.: Robust covariances. In: Gupta, S.S., Moore, D.S. (eds.) Statistical Decision Theory and Related Topics, vol. 11, pp. 165–191. Academic Press, New York (1977)

  13. Kac, M., Kiefer, J., Wolfowitz, J.: On tests of normality and other tests of goodness of fit based on distance methods. Ann. Math. Stat. 26, 189–211 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics, vol. 3, 2nd edn. Griffin, London (1968)

  15. Lee, M.-J.: Nonparametric estimation and test for quadrant correlation in multivariate binary response models. Econ. Rev. 18, 387–415 (1999)

    Article  MATH  Google Scholar 

  16. Li, Y.: Efficient semiparametric regression for longitudinal data with nonparametric covariance estimation. Biometrika 98, 355–370 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Longley, J.W.: An appraisal of least-squares programs from the point of view of the user. J. Am. Stat. Assoc. 62, 819–841 (1967)

    Article  MathSciNet  Google Scholar 

  18. Miller, R.G.: Jackknifing variances. Ann. Math. Stat. 39, 567–582 (1968)

    Article  MATH  Google Scholar 

  19. Neumeyer, N., Dette, H.: A note on one-sided nonparametric analysis of covariance by ranking residuals. Math. Methods Stat. 14, 80–104 (2005)

    MathSciNet  Google Scholar 

  20. Nie, L., Chu, H., Korostyshevskiy, V.R.: Bias reduction for nonparametric correlation coefficients under the bivariate normal copula assumption with known detection limits. Can. J. Stat. 36, 427–442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Owen, A.: Empirical likelihood ratio confidence regions. Ann. Stat. 18, 90–120 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Plachky, D., Rukhin, A.L.: Nonparametric covariance estimation in multivariate distributions. Metrika 50, 131–136 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tsiatis, A.A.: Semiparametric Theory and Missing Data. Springer, New York (2006)

    MATH  Google Scholar 

  24. von Mises, R.: Differentiable statistical functions. Ann. Math. Stat. 18, 309–348 (1947)

    Article  MATH  Google Scholar 

  25. BLINDED CITATION

  26. Wu, W.B., Pourahmadi, M.: Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90, 831–844 (2003)

    Article  MathSciNet  Google Scholar 

  27. Xiao, Z., Linton, O.: A nonparametric prewhitened covariance estimator. J. Time Ser. Anal. 23, 215–250 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Editor and the referee for careful reading and for their comments which greatly improved the paper.

Author information

Authors and Affiliations

  1. Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand

    Christopher S. Withers

  2. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

Authors
  1. Christopher S. Withers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saralees Nadarajah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saralees Nadarajah.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Withers, C.S., Nadarajah, S. Non-parametric confidence intervals for covariance and correlation. METRON 72, 283–306 (2014). https://doi.org/10.1007/s40300-013-0033-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40300-013-0033-9

Keywords

Search

Navigation