The aim of this article is to develop optimal sufficient dimension reduction methodology for the conditional mean in multivariate regression. The context is roughly the same as that of a related method by Cook & Setodji (2003), but the new method has several advantages. It is asymptotically optimal in the sense described herein and its test statistic for dimension always has a chi-squared distribution asymptotically under the null hypothesis. Additionally, the optimal method allows tests of predictor effects. A comparison of the two methods is provided.
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ACKNOWLEDGEMENT The authors are grateful to the referees for many helpful comments. This work was supported in part by grants from the U.S. National Science Foundation.
- Multivariate conditional mean
- Multivariate regression
- Predictor effect test
- Sufficient dimension reduction