Note on distribution-free estimation of maximum linear separation of two multivariate distributions

Abstract

We consider the linear separation of two continuous multivariate distributions. Under mild conditions, the optimal linear separation exists uniquely. A kernel-smoothed approach is proposed to estimate the optimal linear combination and the corresponding separation measure. The proposed method yields consistent estimators allowing the construction of confidence intervals.

Keywords:

• Non-parametric Behrens–Fisher problem
• Measure of separation
• Kernel estimation
• Stress-strength problem
• Receiver operating characteristic curves
• Linear combination

Acknowledgements

The authors would like to thank the Associate Editor and two referees for suggestions that helped very much to improve this article.