EFFICIENT ESTIMATION IN SUFFICIENT DIMENSION REDUCTION.

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.

[1]  A. Tsiatis.  Semiparametric Theory and Missing Data , 2006 .

[2]  Yanyuan Ma,et al.  Constrained local likelihood estimators for semiparametric skew-normal distributions , 2007 .

[3]  Ker-Chau Li,et al.  Sliced Inverse Regression for Dimension Reduction , 1991 .

[4]  S. Weisberg,et al.  Comments on "Sliced inverse regression for dimension reduction" by K. C. Li , 1991 .

[5]  Donglin Zeng,et al.  Efficient Estimation for the Accelerated Failure Time Model , 2007 .

[6]  R. Dennis Cook,et al.  Optimal sufficient dimension reduction in regressions with categorical predictors , 2002 .

[7]  W. Newey,et al.  Semiparametric Efficiency Bounds , 1990 .

[8]  Yu Zhu,et al.  Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression , 2006 .

[9]  Yanyuan Ma,et al.  Efficient semiparametric estimator for heteroscedastic partially linear models , 2006 .

[10]  S. Christian Albright,et al.  Data Analysis and Decision Making with Microsoft Excel , 1999 .

[11]  Marc G. Genton,et al.  Explicit estimating equations for semiparametric generalized linear latent variable models , 2010 .

[12]  Bing Li,et al.  Dimension reduction for non-elliptically distributed predictors: second-order methods , 2010 .

[13]  Shaoli Wang,et al.  On Directional Regression for Dimension Reduction , 2007 .

[14]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[15]  R. Carroll,et al.  Locally Efficient Estimators for Semiparametric Models With Measurement Error , 2006 .

[16]  B. Li,et al.  Dimension reduction for nonelliptically distributed predictors , 2009, 0904.3842.

[17]  Ker-Chau Li,et al.  Regression Analysis Under Link Violation , 1989 .

[18]  Anastasios A. Tsiatis,et al.  Locally efficient semiparametric estimators for functional measurement error models , 2004 .

[19]  Liping Zhu,et al.  Efficient estimation in sufficient dimension reduction , 2013 .

[20]  Ker-Chau Li.  Sliced inverse regression for dimension reduction (with discussion) , 1991 .

[21]  Lixing Zhu,et al.  Dimension Reduction in Regressions Through Cumulative Slicing Estimation , 2010 .

[22]  R. Cook.  On the Interpretation of Regression Plots , 1994 .

[23]  H. Zha,et al.  Contour regression: A general approach to dimension reduction , 2005, math/0508277.

[24]  A. Tsiatis,et al.  Locally Efficient Semiparametric Estimators for Generalized Skew-Elliptical Distributions , 2005 .

[25]  J. Robins,et al.  Estimation of Regression Coefficients When Some Regressors are not Always Observed , 1994 .

[26]  Bing Li,et al.  Groupwise Dimension Reduction , 2010 .

[27]  Liping Zhu,et al.  A Semiparametric Approach to Dimension Reduction , 2012, Journal of the American Statistical Association.

[28]  Y. Xia.  A constructive approach to the estimation of dimension reduction directions , 2007, math/0701761.

[29]  R. Cook.  Regression Graphics , 1994 .

[30]  Jianqing Fan,et al.  Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems , 1996 .