A theoretical note on optimal sufficient dimension reduction with singularity

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In this paper, we discuss an optimal sufficient dimension reduction through minimizing a quadratic objective function proposed by Cook and Ni (2005) with singular inner-product matrix. Within a less restrictive class of inner-product matrices, a generalized inverse of the consistent estimator of the asymptotic covariance matrix gives us the benefit of χ2 statistic and asymptotic efficiency within a subclass.

Original languageEnglish
Pages (from-to)109-113
Number of pages5
JournalStatistics and Probability Letters
StatePublished - 1 Apr 2015

Bibliographical note

Funding Information:
The author is grateful to the associate editor and the referee for many insightful and helpful comments. The author appreciates the provision by the Department of Statistics at the University of Washington of a pleasant research environment in which to initiate this research during his visit in the summer of 2013. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Korean Ministry of Education ( NRF-2014R1A2A1A11049389 ).

Publisher Copyright:
© 2015 Elsevier B.V.


  • Minimum discrepancy approach
  • Singularity
  • Sufficient dimension reduction
  • χ test


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