Generalized principal Hessian directions for mixture multivariate skew elliptical distributions

Principal Hessian directions (pHd) based on the Hessian matrix is a moment-based method and a promising methodology for sufficient dimension reduction because of its easy implementation. However, it requires strong conditions on the distribution of the predictors, which must be nearly Gaussian. We investigate here whether and how this method is applicable when the distribution is a mixture multivariate skew elliptical (MMSE) distribution, and if not, how to adapt the technique. Further, we propose two new estimation algorithms for an extended version of pHd. The theoretical results also serve as a reminder for researchers and users to pay attention to the theoretical conditions on which pHd critically relies. Numerical studies are conducted to examine its performance in finite-sample cases.

[1]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

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

[3]  Lixing Zhu,et al.  Consistently determining the number of factors in multivariate volatility modelling , 2015 .

[4]  Wei Luo.  On the second-order inverse regression methods for a general type of elliptical predictors , 2018 .

[5]  Hsiu J. Ho,et al.  Analysis of multivariate skew normal models with incomplete data , 2009, J. Multivar. Anal..

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

[7]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[8]  Tsung I. Lin,et al.  Maximum likelihood estimation for multivariate skew normal mixture models , 2009, J. Multivar. Anal..

[9]  Tao Wang,et al.  Transformation-based estimation , 2014, Comput. Stat. Data Anal..

[10]  Ker-Chau Li,et al.  Slicing Regression: A Link-Free Regression Method , 1991 .

[11]  Ker-Chau Li,et al.  On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma , 1992 .

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

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

[14]  R. Cook,et al.  Reweighting to Achieve Elliptically Contoured Covariates in Regression , 1994 .

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

[16]  Liping Zhu,et al.  On estimation efficiency of the central mean subspace , 2014 .

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

[18]  R. Dennis Cook,et al.  Dimension Reduction in Regressions With Exponential Family Predictors , 2009 .

[19]  Lixing Zhu,et al.  Sufficient dimension reduction with mixture multivariate skew elliptical distributions , 2017 .

[20]  Trevor Hastie,et al.  Feature Extraction for Nonparametric Discriminant Analysis , 2003 .

[21]  Victor H. Lachos,et al.  Multivariate mixture modeling using skew-normal independent distributions , 2012, Comput. Stat. Data Anal..

[22]  Wei Luo,et al.  Combining eigenvalues and variation of eigenvectors for order determination , 2016 .

[23]  Lixing Zhu,et al.  On Sliced Inverse Regression With High-Dimensional Covariates , 2006 .