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We propose a test of dimension in multivariate regression. This test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model, but it does not require a prespecified model. The test may be particularly useful at the outset of an analysis before a multivariate model is posited, because it can lead to low-dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function.
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Cook et al. (2003) studied this question.
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