Biometrika Advance Access originally published online on April 15, 2008
Biometrika 2008 95(2):469-479; doi:10.1093/biomet/asn002
Articles |
Determining the dimension of the central subspace and central mean subspace
Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, Alabama 36849, U.S.A. zengpen{at}auburn.edu
Received for publication 1 June 2006. Revision received 1 December 2007.
The central subspace and central mean subspace are two important targets of sufficient dimension reduction. We propose a weighted chi-squared test to determine their dimensions based on matrices whose column spaces are exactly equal to the central subspace or the central mean subspace. The asymptotic distribution of the test statistic is obtained. Simulation examples are used to demonstrate the performance of this test.
Key Words: Central mean subspace • Central subspace • Sufficient dimension reduction • Weighted chi-squared test
References
-
Bentler P. M., Xie J. Corrections to test statistics in principal Hessian directions. Statist. Prob. Lett. (2000) 47:381–9.[CrossRef]
Box G. E. P. Some theorems on quadratic forms applied in the study of analysis of variance problems: I. Effect of inequality of variance in one-way classification. Ann. Math. Statist. (1954) 25:290–302.[CrossRef]
Bura E., Cook R. D. Extending sliced inverse regression: the weighted chi-squared test. J. Am. Statist. Assoc. (2001) 96:996–1003.[CrossRef][Web of Science]
Cook R. D. Graphics for regressions with a binary response. J. Am. Statist. Assoc. (1996) 91:983–92.[CrossRef][Web of Science]
Cook R. D. Principal Hessian directions revisited (with Discussion). J. Am. Statist. Assoc. (1998) 93:84–100.[CrossRef][Web of Science]
Cook R. D. Regression Graphics: Ideas For Studying Regressions Through Graphics (1998) New York: John Wiley & Sons.
Cook R. D., Li B. Dimension reduction for conditional mean in regression. Ann. Statist. (2002) 30:455–74.[CrossRef]
Cook R. D., Li B. Determining the dimension of iterative Hessian transformation. Ann. Statist. (2004) 32:2501–31.[CrossRef]
Cook R. D., Nachtsheim C. J. Reweighting to achieve elliptically contoured covariates in regression. J. Am. Statist. Assoc. (1994) 89:592–600.[CrossRef][Web of Science]
Cook R. D., Ni L. Sufficient dimension reduction via inverse regression: A minimum discrepancy approach. J. Am. Statist. Assoc. (2005) 100:410–28.[CrossRef][Web of Science]
Cook R. D., Yin X. Dimension reduction and visualization in discriminant analysis (with Discussion). Aust. New Zeal. J. Statist. (2001) 43:147–99.[CrossRef]
Lee A. J. U-Statistics: Theory and Practice (1990) New York: Marcel Dekker.
Li B., Zha H., Chiaromonte F. Contour regression: a general approach to dimension reduction. Ann. Statist. (2005) 33:1580–1616.[CrossRef]
Li K.-C. Sliced inverse regression for dimension reduction (with Discussion). J. Am. Statist. Assoc. (1991) 86:316–42.[CrossRef][Web of Science]
Li K.-C. On principal Hessian directions for data visualization and dimension reduction: another application of Stein's lemma. J. Am. Statist. Assoc. (1992) 87:1025–39.[CrossRef][Web of Science]
Satterthwaite F. E. Synthesis of variance. Psychometrika (1941) 6:309–16.[CrossRef]
Schott J. R. Determining the dimensionality in sliced inverse regression. J. Am. Statist. Assoc. (1994) 89:141–8.[CrossRef][Web of Science]
Tyler D. E. Asymptotic inference for eigenvectors. Ann. Statist. (1981) 9:725–36.[CrossRef]
Ye Z., Weiss R. E. Using the bootstrap to select one of a new class of dimension reduction methods. J. Am. Statist. Assoc. (2003) 98:968–79.[CrossRef][Web of Science]
Zhu Y., Zeng P. Fourier methods for estimating the central subspace and the central mean subspace in regression. J. Am. Statist. Assoc. (2006) 101:1638–51.[CrossRef][Web of Science]
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||