Skip to main content
SpringerLink
Log in

A new look at measuring dependence

  • Chapter
  • First Online:
Dependence in Probability and Statistics

Part of the book series: Lecture Notes in Statistics ((LNS,volume 200))

Abstract

This paper revisits the concept of dependence.We view statistical dependence as the state of variables being influenced by others. Our viewpoint accords well with the daily understanding of the notion of dependence, while classical dependence measures such as Pearson’s correlation coefficient, Kendall’s τ and Spearman’s ρ have different meanings.With this understanding of dependence, we introduce new dependence measures which are easy to work with and they are useful for developing an asymptotic theory for complicated stochastic systems. We also explore relations of the introduced dependence concept with nonlinear system theory, experimental design, information theory and risk management.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arjas, E. and Lehtonen, T. (1978) Approximating many server queues by means of single server queues. Math. Operation Research 3, 205-223.

    Article  MATH  MathSciNet  Google Scholar 

  2. Barlow, R. and Proschan, F. (1975) Statistical theory of reliability and life testing, Rinehart and Winston, New York.

    MATH  Google Scholar 

  3. Bjerve S. and Doksum, K. (1993) Correlation curves: measures of association as functions of covariates values. Annals of Statistics 21, 890-902.

    Article  MATH  MathSciNet  Google Scholar 

  4. Blyth, S. (1994) Local divergence and association. Biometrika 81, 579–584.

    Article  MATH  MathSciNet  Google Scholar 

  5. Block, H.W., Sampson A. R. and Savits, T. H. (1990) (eds) Topics in statistical dependence, IMS Lecture Notes-Monograph Series, 16 IMS, Hayward, CA.

    Google Scholar 

  6. Borkar, V. S. (1993) White-noise representations in stochastic realization theory. SIAM J. Control Optim. 31 1093–1102.

    Article  MATH  MathSciNet  Google Scholar 

  7. Bradley, R. (2005) Basic properties of strong mixing conditions. A Survey and some open questions. Probability Surveys 2, 107-144

    Article  MATH  MathSciNet  Google Scholar 

  8. Bryk, A. and Mielniczuk, J. (2005) Asymptotic properties of kernel density estimates for linear processes: application of projection method. Nonparametric Statistics 14, 121-133.

    Article  MathSciNet  Google Scholar 

  9. Capéraà, P. and Genest, C. (1990). Concepts de dépendance et ordres stochastiques pour des lois bidimensionelles. Canadian Journal of Statistics 18, 315-326.

    MATH  Google Scholar 

  10. Cowell, R. G., Dawid, A. P. and Spiegelhalter, D. J. (1999) Probabilistic Networks and Systems, Springer, New York.

    MATH  Google Scholar 

  11. Crutchfeld, J.P. and Feldman, D.P. (2003) Regularities unseen, randomness observed: The entropy convergence hierarchy. Chaos 15, 25-54.

    Article  Google Scholar 

  12. Deák, I. (1990) Random Numbers Generators and Simulation, Akadémiai Kiadó, Budapest.

    Google Scholar 

  13. Dedecker, J., Merlevède, F. (2002) Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30, 1044–1081.

    Article  MATH  MathSciNet  Google Scholar 

  14. Dedecker, J., P. Doukhan, G. Lang, J.R. Leon R. S. Louhichi and C. Prieur (2007) Weak Dependence: With Examples and Applications, Springer, New York.

    MATH  Google Scholar 

  15. Diaconis, P. and Freedman, D. (1999) Iterated random functions. SIAM Review 41 41-76.

    Article  Google Scholar 

  16. Doksum, K. and Froda, S. M. (2000) Neighborhood correlation. J. Statist. Plan. Inf. 91, 267–294.

    Article  MATH  MathSciNet  Google Scholar 

  17. Doksum, K. and Samarov, A. (1995) Nonparametric estimation of global functionals and measure of the explanatory powers in regression. Annals of Statistics 23, 1443-1473.

    Article  MATH  MathSciNet  Google Scholar 

  18. Doukhan, P. and Louhichi, S. (1999) A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313–342.

    Article  MATH  MathSciNet  Google Scholar 

  19. Drouet Mari, D. and Kotz, S. (2001) Correlation and dependence, Imperial College Press, London.

    Book  MATH  Google Scholar 

  20. Efron, B. and Stein, C. (1981) The jackknife estimate of variance. Annals of Statistics 9, 586–596.

    Article  MATH  MathSciNet  Google Scholar 

  21. Embrechts, P., McNeil, A. and Straumann, D. (2002) Correlatation and dependence in risk management: properties and pitfalls. In: RiskManagement: Value at Risk and Beyond. (M.A.H. Dempster, ed.), Cambridge University Press, 176-223.

    Chapter  Google Scholar 

  22. Esary, J. D., Proschan, F. and Walkup, D.W. (1967) Association of random variables with applications, Annals of Mathematical Statistics 38, 1466–1474.

    Article  MATH  MathSciNet  Google Scholar 

  23. Esary, J. D. and Proschan, F. (1972) Relationships between some concepts of bivariate dependence. Annals of Mathematical Statistics 43, 651-655.

    Article  MATH  MathSciNet  Google Scholar 

  24. Glymour, C. (2006) Causation-II. Encyclopedia of Statistical Sciences, Wiley. 782–794

    Google Scholar 

  25. Gordin, M.I. (1969) The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741

    MathSciNet  Google Scholar 

  26. Gordin, M.I. and Lifsic, B.A.(1973) The central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239, 392–393

    MathSciNet  Google Scholar 

  27. Granger, C.W.J. (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.

    Article  Google Scholar 

  28. Hannan, E.J. (1979) The central limit theorem for time series regression. Stochastic Processes Appl., 9, 281–289

    Article  MATH  MathSciNet  Google Scholar 

  29. Joe, H. (1997) Multivariate models and dependence concepts, Chapman and Hall, London.

    MATH  Google Scholar 

  30. Jogdeo, K. (1978) On a probability bound of Marshall and Olkin. Ann. Statist. 6, 232–234.

    Article  MATH  MathSciNet  Google Scholar 

  31. Jogdeo, K. (2006) Concepts of dependence. Encyclopedia of Statistical Sciences, Wiley, 163–1647.

    Google Scholar 

  32. Kalikow, S. A. (1982) T, T−1 transformation is not loosely Bernoulli. Ann. Math. 115, 393–409.

    Article  MathSciNet  Google Scholar 

  33. Kallianpur, G. (1981) Some ramifications of Wieners ideas on nonlinear prediction. In: Norbert Wiener, Collected Works with Commentaries, MIT Press, Mass., 402–424.

    Google Scholar 

  34. Lai, C. D. and Xie, M. (2006) Stochastic Ageing and Dependence for Reliability. Springer, New York.

    MATH  Google Scholar 

  35. Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.

    Article  MATH  MathSciNet  Google Scholar 

  36. Li, H., Scarsini, M. and Shaked, M. (1996) Linkages: A tool for construction of multivariate distributions with nonoverlapping multivariate marginals. J. Multivariate. Anal. 56, 20–41.

    Article  MATH  MathSciNet  Google Scholar 

  37. Liggett, T.M. (1985) Interacting Particle Systems, Springer, New York

    MATH  Google Scholar 

  38. Mellor, D. H. (1998) The Facts of Causation, Routledge, New York.

    Google Scholar 

  39. Merlevède F., and Peligrad, M. (2006) On the weak invariance principle for stationary sequences under projective criteria. J. Theor. Probab. 19, 647–689

    Article  MATH  Google Scholar 

  40. J. P. Morgan (1996) RiskMetrics. Technical Document. New York.

    Google Scholar 

  41. Mosteller, F. and Tukey, J.W. (1977) Data Analysis and Regression, Addison-Wesley Reading, Mass.

    Google Scholar 

  42. Newman, C.M. (1984) Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Inequalities in Statistics and Probability (Y.L. Tong, ed.), IMS Lecture Notes - Monograph Series 7, 127–140.

    Google Scholar 

  43. Ornstein, D. S. (1973) An example of a Kolmogorov automorphism that is not a Bernoulli shift. Advances in Math. 10 49–62

    Article  MATH  MathSciNet  Google Scholar 

  44. Rényi, A. (1959) On measures of dependence. Acta Math. Acad. Sci. Hungar. 10 441–451

    Article  MATH  MathSciNet  Google Scholar 

  45. Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23 470–472.

    Article  MATH  MathSciNet  Google Scholar 

  46. Rosenblatt, M. (1959). Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8, 665–681.

    MATH  MathSciNet  Google Scholar 

  47. Rosenblatt, M. (1971) Markov Processes. Structure and Asymptotic Behavior, Springer, New York.

    Google Scholar 

  48. Rosenblatt, M. (2009) A comment on a conjecture of N. Wiener. Statist. Probab. Letters, 79, 347–348

    Article  MATH  MathSciNet  Google Scholar 

  49. Rubin, D. (1990) Formal modes of statistical inference for causal effects. Journal of Statistical Planning and Inference 25, 279–292.

    Article  Google Scholar 

  50. Rüschendorf, L. (1981) Stochastically ordered distributions and monotonicity of the OC-function of sequential probability ratio tests. Mathematische Operationsforschung und Statistik. Series Statistcis 12, 327–338.

    MATH  Google Scholar 

  51. Rüschendorf, L. and de Valk, V.(1981) On regression representation of stochastic processes. Stochastic Processes and Their Applications 46, 183–198.

    Article  Google Scholar 

  52. Scarsini, M. (1984) On measures of concordance. Stochastica 8, 201–218.

    MATH  MathSciNet  Google Scholar 

  53. Skaug, H. J. and Tjøstheim, D. (1993) A nonparametric test of serial independence based on the empirical distribution function. Biometrika 80 591–602.

    Article  MATH  MathSciNet  Google Scholar 

  54. Stigler, S. M. (1986) The history of statistics, Harvard University Press, Cambridge, MA.

    MATH  Google Scholar 

  55. Szekli, R. (1995) Stochastic ordering and dependence in applied probability, Springer, New York

    MATH  Google Scholar 

  56. Tjøstheim, D. (1996) Measures of dependence and tests of independence. Statistics 28 249–284.

    Article  MathSciNet  Google Scholar 

  57. Tong, H. (1990) Nonlinear time series analysis: A dynamic approach., Oxford University Press, Oxford.

    Google Scholar 

  58. Vol?ny, D. (1993) Approximating martingales and the central limit theorem for strictly stationary processes. Stochastic Processes and their Applications 44, 41–74

    Article  MathSciNet  Google Scholar 

  59. Wiener, N. (1958) Nonlinear Problems in Random Theory., MIT Press, Cambridge, MA.

    Google Scholar 

  60. Woodroofe, M. (1992) A central limit theorem for functions of aMarkov chain with applications to shifts. Stochastic Processes and Their Applications 41, 33–44.

    Article  MATH  MathSciNet  Google Scholar 

  61. Wu, W.B. and Woodroofe, M. (2004) Martingale approximations for sums of stationary processes. Ann. Probab. 32, 1674–1690.

    Article  MATH  MathSciNet  Google Scholar 

  62. Wu, W. B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences, USA, 102, 14150–14154.

    Google Scholar 

  63. Wu, W. B. (2007) Strong invariance principles for dependent randomvariables. Ann. Probab., 35, 2294–2320.

    Article  MATH  MathSciNet  Google Scholar 

  64. Wu, W. B. (2008) Empirical processes of stationary sequences. Statistica Sinica, 18 313–333.

    MATH  MathSciNet  Google Scholar 

  65. Wu, W. B. and Shao, X. (2004) Limit theorems for iterated random functions. Journal of Applied Probability, 41 425–436.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, The University of Chicago, University of Chicago, Chicago, IL, 60637, USA

    Wei Biao Wu

  2. Polish Academy of Sciences, Warsaw University of Technology, Warsaw, Poland

    Jan Mielniczuk

Authors
  1. Wei Biao Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Mielniczuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Biao Wu .

Editor information

Editors and Affiliations

  1. UFR Sciences-Techniques, avenue Adolphe-Chauvin 2, Pontoise, 95302, France

    Paul Doukhan

  2. , UMR MIA 518, INRA AgroParisTech, Paris, 75005, France

    Gabriel Lang

  3. Stochastic Processes Department, Akademijos str. 4, Vilnius, 08412, Lithuania

    Donatas Surgailis

  4. Aarhus University, School of Economics, CREATES, Bartholins Allé 10, Aarhus C, 8000, Denmark

    Gilles Teyssière

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Wu, W.B., Mielniczuk, J. (2010). A new look at measuring dependence. In: Doukhan, P., Lang, G., Surgailis, D., Teyssière, G. (eds) Dependence in Probability and Statistics. Lecture Notes in Statistics(), vol 200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14104-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation