Annals of Statistics

Efficient estimation in sufficient dimension reduction

Yanyuan Ma and Liping Zhu

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Abstract

We develop an efficient estimation procedure for identifying and estimating the central subspace. Using a new way of parameterization, we convert the problem of identifying the central subspace to the problem of estimating a finite dimensional parameter in a semiparametric model. This conversion allows us to derive an efficient estimator which reaches the optimal semiparametric efficiency bound. The resulting efficient estimator can exhaustively estimate the central subspace without imposing any distributional assumptions. Our proposed efficient estimation also provides a possibility for making inference of parameters that uniquely identify the central subspace. We conduct simulation studies and a real data analysis to demonstrate the finite sample performance in comparison with several existing methods.

Article information

Source
Ann. Statist., Volume 41, Number 1 (2013), 250-268.

Dates
First available in Project Euclid: 26 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1364302742

Digital Object Identifier
doi:10.1214/12-AOS1072

Mathematical Reviews number (MathSciNet)
MR3059417

Zentralblatt MATH identifier
1347.62089

Subjects
Primary: 62H12: Estimation 62J02: General nonlinear regression
Secondary: 62F12: Asymptotic properties of estimators

Keywords
Central subspace dimension reduction estimating equations semiparametric efficiency sliced inverse regression

Citation

Ma, Yanyuan; Zhu, Liping. Efficient estimation in sufficient dimension reduction. Ann. Statist. 41 (2013), no. 1, 250--268. doi:10.1214/12-AOS1072. https://projecteuclid.org/euclid.aos/1364302742


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References

  • [1] Albright, S. C., Winston, W. L. and Zappe, C. J. (1999). Data Analysis and Decision Making with Microsoft Excel. Duxbury, Pacific Grove, CA.
  • [2] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD.
    Mathematical Reviews (MathSciNet): MR1245941
  • [3] Chiaromonte, F., Cook, R. D. and Li, B. (2002). Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30 475–497.
    Mathematical Reviews (MathSciNet): MR1902896
    Zentralblatt MATH: 1012.62036
    Digital Object Identifier: doi:10.1214/aos/1021379862
    Project Euclid: euclid.aos/1021379862
  • [4] Cook, R. D. (1994). On the interpretation of regression plots. J. Amer. Statist. Assoc. 89 177–189.
    Mathematical Reviews (MathSciNet): MR1266295
    Zentralblatt MATH: 0791.62066
    Digital Object Identifier: doi:10.1080/01621459.1994.10476459
  • [5] Cook, R. D. (1998). Regression Graphics. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1645673
  • [6] Cook, R. D. and Weisberg, S. (1991). Comment on “Sliced inverse regression for dimension reduction,” by K.-C. Li. J. Amer. Statist. Assoc. 86 328–332.
    Mathematical Reviews (MathSciNet): MR1137117
    Zentralblatt MATH: 0742.62044
    Digital Object Identifier: doi:10.1080/01621459.1991.10475035
  • [7] Dong, Y. and Li, B. (2010). Dimension reduction for non-elliptically distributed predictors: Second-order methods. Biometrika 97 279–294.
    Mathematical Reviews (MathSciNet): MR2650738
    Zentralblatt MATH: 1233.62119
    Digital Object Identifier: doi:10.1093/biomet/asq016
  • [8] Fan, J., Yao, Q. and Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83 189–206.
    Mathematical Reviews (MathSciNet): MR1399164
    Zentralblatt MATH: 0865.62026
    Digital Object Identifier: doi:10.1093/biomet/83.1.189
  • [9] Li, B. and Dong, Y. (2009). Dimension reduction for nonelliptically distributed predictors. Ann. Statist. 37 1272–1298.
    Mathematical Reviews (MathSciNet): MR2509074
    Zentralblatt MATH: 1160.62050
    Digital Object Identifier: doi:10.1214/08-AOS598
    Project Euclid: euclid.aos/1239369022
  • [10] Li, B. and Wang, S. (2007). On directional regression for dimension reduction. J. Amer. Statist. Assoc. 102 997–1008.
    Mathematical Reviews (MathSciNet): MR2354409
    Zentralblatt MATH: 05564427
    Digital Object Identifier: doi:10.1198/016214507000000536
  • [11] Li, B., Zha, H. and Chiaromonte, F. (2005). Contour regression: A general approach to dimension reduction. Ann. Statist. 33 1580–1616.
    Mathematical Reviews (MathSciNet): MR2166556
    Zentralblatt MATH: 1078.62033
    Digital Object Identifier: doi:10.1214/009053605000000192
    Project Euclid: euclid.aos/1123250223
  • [12] Li, K.-C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316–342.
    Mathematical Reviews (MathSciNet): MR1137117
    Zentralblatt MATH: 0742.62044
    Digital Object Identifier: doi:10.1080/01621459.1991.10475035
  • [13] Li, K.-C. and Duan, N. (1989). Regression analysis under link violation. Ann. Statist. 17 1009–1052.
    Mathematical Reviews (MathSciNet): MR1015136
    Zentralblatt MATH: 0753.62041
    Digital Object Identifier: doi:10.1214/aos/1176347254
    Project Euclid: euclid.aos/1176347254
  • [14] Li, L., Li, B. and Zhu, L.-X. (2010). Groupwise dimension reduction. J. Amer. Statist. Assoc. 105 1188–1201.
    Mathematical Reviews (MathSciNet): MR2752614
    Digital Object Identifier: doi:10.1198/jasa.2010.tm09643
  • [15] Ma, Y. and Carroll, R. J. (2006). Locally efficient estimators for semiparametric models with measurement error. J. Amer. Statist. Assoc. 101 1465–1474.
    Mathematical Reviews (MathSciNet): MR2279472
    Zentralblatt MATH: 1171.62324
    Digital Object Identifier: doi:10.1198/016214506000000519
  • [16] Ma, Y., Chiou, J.-M. and Wang, N. (2006). Efficient semiparametric estimator for heteroscedastic partially linear models. Biometrika 93 75–84.
    Mathematical Reviews (MathSciNet): MR2277741
    Zentralblatt MATH: 1152.62020
    Digital Object Identifier: doi:10.1093/biomet/93.1.75
  • [17] Ma, Y. and Genton, M. G. (2010). Explicit estimating equations for semiparametric generalized linear latent variable models. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 475–495.
    Mathematical Reviews (MathSciNet): MR2758524
    Digital Object Identifier: doi:10.1111/j.1467-9868.2010.00741.x
  • [18] Ma, Y., Genton, M. G. and Tsiatis, A. A. (2005). Locally efficient semiparametric estimators for generalized skew-elliptical distributions. J. Amer. Statist. Assoc. 100 980–989.
    Mathematical Reviews (MathSciNet): MR2201024
    Zentralblatt MATH: 1117.62394
    Digital Object Identifier: doi:10.1198/016214505000000079
  • [19] Ma, Y. and Hart, J. D. (2007). Constrained local likelihood estimators for semiparametric skew-normal distributions. Biometrika 94 119–134.
    Mathematical Reviews (MathSciNet): MR2307902
    Zentralblatt MATH: 1143.62019
    Digital Object Identifier: doi:10.1093/biomet/asm020
  • [20] Ma, Y. and Zhu, L. (2012). A semiparametric approach to dimension reduction. J. Amer. Statist. Assoc. 107 168–179.
    Mathematical Reviews (MathSciNet): MR2949349
    Digital Object Identifier: doi:10.1080/01621459.2011.646925
  • [21] Ma, Y. and Zhu, L. (2013). Supplement to “Efficient estimation in sufficient dimension reduction.” DOI:10.1214/12-AOS1072SUPP.
  • [22] Newey, W. (1990). Semiparametric efficiency bounds. J. Appl. Econometrics 5 99–135.
  • [23] Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. J. Amer. Statist. Assoc. 89 846–866.
    Mathematical Reviews (MathSciNet): MR1294730
    Zentralblatt MATH: 0815.62043
    Digital Object Identifier: doi:10.1080/01621459.1994.10476818
  • [24] Tsiatis, A. A. (2006). Semiparametric Theory and Missing Data. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2233926
    Zentralblatt MATH: 1105.62002
  • [25] Tsiatis, A. A. and Ma, Y. (2004). Locally efficient semiparametric estimators for functional measurement error models. Biometrika 91 835–848.
    Mathematical Reviews (MathSciNet): MR2126036
    Zentralblatt MATH: 1064.62034
    Digital Object Identifier: doi:10.1093/biomet/91.4.835
  • [26] Xia, Y. (2007). A constructive approach to the estimation of dimension reduction directions. Ann. Statist. 35 2654–2690.
    Mathematical Reviews (MathSciNet): MR2382662
    Zentralblatt MATH: 05241119
    Digital Object Identifier: doi:10.1214/009053607000000352
    Project Euclid: euclid.aos/1201012976
  • [27] Zeng, D. and Lin, D. Y. (2007). Efficient estimation in the accelerated failure time model. J. Amer. Statist. Assoc. 102 1387–1396.
    Mathematical Reviews (MathSciNet): MR2412556
    Zentralblatt MATH: 05564458
    Digital Object Identifier: doi:10.1198/016214507000001085
  • [28] Zeng, D. and Lin, D. Y. (2007). Maximum likelihood estimation in semiparametric models with censored data (with discussion). J. Roy. Statist. Soc. Ser. B 69 507–564.
    Mathematical Reviews (MathSciNet): MR2370068
    Digital Object Identifier: doi:10.1111/j.1369-7412.2007.00606.x
  • [29] Zhu, L.-P., Zhu, L.-X. and Feng, Z.-H. (2010). Dimension reduction in regressions through cumulative slicing estimation. J. Amer. Statist. Assoc. 105 1455–1466.
    Mathematical Reviews (MathSciNet): MR2796563
    Digital Object Identifier: doi:10.1198/jasa.2010.tm09666
  • [30] Zhu, Y. and Zeng, P. (2006). Fourier methods for estimating the central subspace and the central mean subspace in regression. J. Amer. Statist. Assoc. 101 1638–1651.
    Mathematical Reviews (MathSciNet): MR2279485
    Zentralblatt MATH: 1171.62325
    Digital Object Identifier: doi:10.1198/016214506000000140

Supplemental materials

  • Supplementary material: Supplement to “Efficient estimation in sufficient dimension reduction”. The supplement file aos1072_supp.pdf is available upon request. It contains derivations of the efficient score for model (2.1) and an outline of proof for Theorems 1 and 2.
    Digital Object Identifier: doi:10.1214/12-AOS1072SUPP
    Supplemental PDF (254 KB)

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