Weighted Distance Weighted Discrimination and Its Asymptotic Properties

Xingye Qiao Xingye Qiao is Ph.D. Student, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599 . Hao Helen Zhang is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695. Yufeng Liu is Associate Professor, Department of Statistics and Operations Research, Carolina Center for Genome Sciences, University of North Carolina, Chapel Hill, NC 27599. Michael J. Todd is Professor, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853. J. S. Marron is Professor, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599. We thank the Editor, the Associate Editor, and four referees for many helpful suggestions that lead to improvement of the paper. The research of Qiao and Marron was supported in part by NSF grant DMS-0606577, and NIH grants R01-05-010134 and P50-CA58223-15. The research of Liu was supported in part by NSF grants DMS-0606577 and DMS-0747575, and NIH/NCI grant R01-CA-149569. The research of Zhang was supported in part by NSF grant DMS-0645293 and NIH/NCI grant R01-CA-085848. The research of Todd was supported in part by NSF grant DMS-0513337 and ONR grant N00014-02-1-0057.

Hao Helen Zhang Xingye Qiao is Ph.D. Student, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599 . Hao Helen Zhang is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695. Yufeng Liu is Associate Professor, Department of Statistics and Operations Research, Carolina Center for Genome Sciences, University of North Carolina, Chapel Hill, NC 27599. Michael J. Todd is Professor, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853. J. S. Marron is Professor, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599. We thank the Editor, the Associate Editor, and four referees for many helpful suggestions that lead to improvement of the paper. The research of Qiao and Marron was supported in part by NSF grant DMS-0606577, and NIH grants R01-05-010134 and P50-CA58223-15. The research of Liu was supported in part by NSF grants DMS-0606577 and DMS-0747575, and NIH/NCI grant R01-CA-149569. The research of Zhang was supported in part by NSF grant DMS-0645293 and NIH/NCI grant R01-CA-085848. The research of Todd was supported in part by NSF grant DMS-0513337 and ONR grant N00014-02-1-0057.

Yufeng Liu Xingye Qiao is Ph.D. Student, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599 . Hao Helen Zhang is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695. Yufeng Liu is Associate Professor, Department of Statistics and Operations Research, Carolina Center for Genome Sciences, University of North Carolina, Chapel Hill, NC 27599. Michael J. Todd is Professor, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853. J. S. Marron is Professor, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599. We thank the Editor, the Associate Editor, and four referees for many helpful suggestions that lead to improvement of the paper. The research of Qiao and Marron was supported in part by NSF grant DMS-0606577, and NIH grants R01-05-010134 and P50-CA58223-15. The research of Liu was supported in part by NSF grants DMS-0606577 and DMS-0747575, and NIH/NCI grant R01-CA-149569. The research of Zhang was supported in part by NSF grant DMS-0645293 and NIH/NCI grant R01-CA-085848. The research of Todd was supported in part by NSF grant DMS-0513337 and ONR grant N00014-02-1-0057.

Michael J. Todd Xingye Qiao is Ph.D. Student, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599 . Hao Helen Zhang is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695. Yufeng Liu is Associate Professor, Department of Statistics and Operations Research, Carolina Center for Genome Sciences, University of North Carolina, Chapel Hill, NC 27599. Michael J. Todd is Professor, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853. J. S. Marron is Professor, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599. We thank the Editor, the Associate Editor, and four referees for many helpful suggestions that lead to improvement of the paper. The research of Qiao and Marron was supported in part by NSF grant DMS-0606577, and NIH grants R01-05-010134 and P50-CA58223-15. The research of Liu was supported in part by NSF grants DMS-0606577 and DMS-0747575, and NIH/NCI grant R01-CA-149569. The research of Zhang was supported in part by NSF grant DMS-0645293 and NIH/NCI grant R01-CA-085848. The research of Todd was supported in part by NSF grant DMS-0513337 and ONR grant N00014-02-1-0057.

J. S. Marron Xingye Qiao is Ph.D. Student, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599 . Hao Helen Zhang is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695. Yufeng Liu is Associate Professor, Department of Statistics and Operations Research, Carolina Center for Genome Sciences, University of North Carolina, Chapel Hill, NC 27599. Michael J. Todd is Professor, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853. J. S. Marron is Professor, Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599. We thank the Editor, the Associate Editor, and four referees for many helpful suggestions that lead to improvement of the paper. The research of Qiao and Marron was supported in part by NSF grant DMS-0606577, and NIH grants R01-05-010134 and P50-CA58223-15. The research of Liu was supported in part by NSF grants DMS-0606577 and DMS-0747575, and NIH/NCI grant R01-CA-149569. The research of Zhang was supported in part by NSF grant DMS-0645293 and NIH/NCI grant R01-CA-085848. The research of Todd was supported in part by NSF grant DMS-0513337 and ONR grant N00014-02-1-0057.

Abstract

While Distance Weighted Discrimination (DWD) is an appealing approach to classification in high dimensions, it was designed for balanced datasets. In the case of unequal costs, biased sampling, or unbalanced data, there are major improvements available, using appropriately weighted versions of DWD (wDWD). A major contribution of this paper is the development of optimal weighting schemes for various nonstandard classification problems. In addition, we discuss several alternative criteria and propose an adaptive weighting scheme (awDWD) and demonstrate its advantages over nonadaptive weighting schemes under some situations. The second major contribution is a theoretical study of weighted DWD. Both high-dimensional low sample-size asymptotics and Fisher consistency of DWD are studied. The performance of weighted DWD is evaluated using simulated examples and two real data examples. The theoretical results are also confirmed by simulations.

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Weighted Distance Weighted Discrimination and Its Asymptotic Properties

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