The Annals of Statistics

Testing conditional independence using maximal nonlinear conditional correlation

Tzee-Ming Huang

Full-text: Open access


In this paper, the maximal nonlinear conditional correlation of two random vectors X and Y given another random vector Z, denoted by ρ1(X, Y|Z), is defined as a measure of conditional association, which satisfies certain desirable properties. When Z is continuous, a test for testing the conditional independence of X and Y given Z is constructed based on the estimator of a weighted average of the form ∑k=1nZfZ(zk)ρ12(X, Y|Z = zk), where fZ is the probability density function of Z and the zk’s are some points in the range of Z. Under some conditions, it is shown that the test statistic is asymptotically normal under conditional independence, and the test is consistent.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 2047-2091.

First available in Project Euclid: 11 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 62H15: Hypothesis testing 62G10: Hypothesis testing

Measure of association measure of conditional association conditional independence test


Huang, Tzee-Ming. Testing conditional independence using maximal nonlinear conditional correlation. Ann. Statist. 38 (2010), no. 4, 2047--2091. doi:10.1214/09-AOS770.

Export citation


  • [1] Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29–54.
  • [2] Daudin, J.-J. (1980). Partial association measures and an application to qualitative regression. Biometrika 67 581–590.
  • [3] Dauxois, J. and Nkiet, G. M. (1998). Nonlinear canonical analysis and independence tests. Ann. Statist. 26 1254–1278.
  • [4] Dauxois, J. and Nkiet, G. M. (2002). Measures of association for Hilbertian subspaces and some applications. J. Multivariate Anal. 82 263–298.
  • [5] Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.
  • [6] Hsing, T., Liu, L.-Y., Brun, M. and Dougherty, E. R. (2005). The coefficient of intrinsic dependence. Pattern Recognition 38 623–636.
  • [7] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • [8] Paparoditis, E. and Politis, D. N. (2000). The local bootstrap for kernel estimators under general dependence conditions. Ann. Inst. Statist. Math. 52 139–159.
  • [9] Rényi, A. (1959). On measures of dependence. Acta Math. Acad. Sci. Hungar. 10 441–451.
  • [10] Romanovič, V. A. (1975). The maximal partial correlation coefficient of two σ-algebras relative to a third σ-algebra. Izv. Vysš. Učebn. Zaved. Matematika 10 94–96.
  • [11] Schumaker, L. L. (1981). Spline Functions. Wiley, New York.
  • [12] Su, L. and White, H. (2007). A consistent characteristic function-based test for conditional independence. J. Econometrics 141 807–834.
  • [13] Su, L. and White, H. (2008). A nonparametric Hellinger metric test for conditional independence. Econometric Theory 24 829–864.