×

zbMATH — the first resource for mathematics

Asymptotics for sliced average variance estimation. (English) Zbl 1114.62053
Summary: We systematically study consistency of sliced average variance estimation (SAVE). The findings reveal that when the response is continuous, the asymptotic behavior of SAVE is rather different from that of sliced inverse regression (SIR). SIR can achieve \(\sqrt{n}\)-consistency even when each slice contains only two data points. However, SAVE cannot be \(\sqrt{n}\)-consistent and it even turns out to be not consistent when each slice contains a fixed number of data points that do not depend on \(n\), where \(n\) is the sample size. These results theoretically confirm the notion that SAVE is more sensitive to the number of slices than SIR. Taking this into account, a bias correction is recommended in order to allow SAVE to be \(\sqrt{n}\)-consistent. In contrast, when the response is discrete and takes finite values, \(\sqrt{n}\)-consistency can be achieved. Therefore, an approximation through discretization, which is commonly used in practice, is studied. A simulation study is carried out for the purposes of illustration.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
65C60 Computational problems in statistics (MSC2010)
62E20 Asymptotic distribution theory in statistics
62H99 Multivariate analysis
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80 580–619. JSTOR: · Zbl 0594.62044 · doi:10.2307/2288473 · links.jstor.org
[2] Chen, X., Fang, Z., Li, G. Y. and Tao, B. (1989). Nonparametric Statistics . Shanghai Science and Technology Press, Shanghai. (In Chinese.)
[3] Cook, R. D. (1994). On the interpretation of regression plots. J. Amer. Statist. Assoc. 89 177–189. JSTOR: · Zbl 0791.62066 · doi:10.2307/2291214 · links.jstor.org
[4] Cook, R. D. (1998). Regression Graphics: Ideas for Studying Regressions through Graphics . Wiley, New York. · Zbl 0903.62001
[5] Cook, R. D. (2000). SAVE: A method for dimension reduction and graphics in regression. Comm. Statist. Theory Methods 29 2109–2121. · Zbl 1061.62503 · doi:10.1080/03610920008832598
[6] Cook, R. D. and Critchley, F. (2000). Identifying regression outliers and mixtures graphically. J. Amer. Statist. Assoc. 95 781–794. JSTOR: · Zbl 0999.62056 · doi:10.2307/2669462 · links.jstor.org
[7] Cook, R. D. and Li, B. (2002). Dimension reduction for conditional mean in regression. Ann. Statist. 30 455–474. · Zbl 1012.62035 · doi:10.1214/aos/1021379861
[8] Cook, R. D. and Ni, L. (2005). Sufficient dimension reduction via inverse regression: A minimum discrepancy approach. J. Amer. Statist. Assoc. 100 410–428. · Zbl 1117.62312 · doi:10.1198/016214504000001501 · miranda.asa.catchword.org
[9] Cook, R. D. and Weisberg, S. (1991). Discussion of “Sliced inverse regression for dimension reduction,” by K.-C. Li. J. Amer. Statist. Assoc. 86 328–332. JSTOR: · Zbl 0742.62044 · doi:10.2307/2290563 · links.jstor.org
[10] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods . Springer, New York. · Zbl 1014.62103
[11] Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression, J. Amer. Statist. Assoc. 76 817–823. JSTOR: · doi:10.2307/2287576 · links.jstor.org
[12] Gannoun, A. and Saracco, J. (2003). An asymptotic theory for SIR\(_\alpha\) method. Statist. Sinica 13 297–310. · Zbl 1015.62036
[13] Hooper, J. (1959). Simultaneous equations and canonical correlation theory. Econometrica 27 245–256. JSTOR: · Zbl 0087.15302 · doi:10.2307/1909445 · links.jstor.org
[14] Hsing, T. and Carroll, R. J. (1992). An asymptotic theory for sliced inverse regression. Ann. Statist. 20 1040–1061. · Zbl 0821.62019 · doi:10.1214/aos/1176348669
[15] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd ed. Springer, New York. · Zbl 0996.60001
[16] Li, K.-C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316–342. JSTOR: · Zbl 0742.62044 · doi:10.2307/2290563 · links.jstor.org
[17] Li, K.-C. (1992). On principal Hessian directions for data visualization and dimension reduction: Another application of Stein’s lemma. J. Amer. Statist. Assoc. 87 1025–1039. JSTOR: · Zbl 0765.62003 · doi:10.2307/2290640 · links.jstor.org
[18] Li, Y. X. and Zhu, L.-X. (2005). Asymptotics for sliced average variance estimation. Technical Report, Dept. Mathematics, Hong Kong Baptist Univ.
[19] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, New York. · Zbl 0544.60045
[20] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002
[21] Stute, W., Thies, S. and Zhu, L.-X. (1998). Model checks for regression: An innovation process approach. Ann. Statist. 26 1916–1934. · Zbl 0930.62044 · doi:10.1214/aos/1024691363
[22] Stute, W. and Zhu, L.-X. (2005). Nonparametric checks for single-index models. Ann. Statist. 33 1048–1083. · Zbl 1080.62023 · doi:10.1214/009053605000000020
[23] Xia, Y., Tong, H., Li, W. K. and Zhu, L.-X. (2002). An adaptive estimation of dimension reduction space. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 363–410. JSTOR: · Zbl 1091.62028 · doi:10.1111/1467-9868.03411 · links.jstor.org
[24] Ye, Z. and Weiss, R. E. (2003). Using the bootstrap to select one of a new class of dimension-reduction methods. J. Amer. Statist. Assoc. 98 968–979. · Zbl 1045.62034 · doi:10.1198/016214503000000927
[25] Zhu, L.-X. and Fang, K.-T. (1996). Asymptotics for kernel estimate of sliced inverse regression. Ann. Statist. 24 1053–1068. · Zbl 0864.62027 · doi:10.1214/aos/1032526955
[26] Zhu, L.-X., Miao, B. and Peng, H. (2006). On sliced inverse regression with high-dimensional covariates. J. Amer. Statist. Assoc. 101 630–643. · Zbl 1119.62331 · doi:10.1198/016214505000001285 · miranda.asa.catchword.org
[27] Zhu, L.-X. and Ng, K. W. (1995). Asymptotics of sliced inverse regression. Statist. Sinica 5 727–736. · Zbl 0824.62036
[28] Zhu, L.-X., Ohtaki, M. and Li, Y. X. (2007). On hybrid methods of inverse regression-based algorithms. Comput. Statist. Data Anal. 51 2621–2635. · Zbl 1161.62332 · doi:10.1016/j.csda.2006.01.005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.