On testing common indices for several multi-index models: A link-free approach
"To avoid the curse of dimensionality, and to help us better understand the structure of the high dimensional data, methods for dimension reduction are clearly called for. The common linear dimension reduction techniques for single population include principal component analysis (PCA) which is unsupervised in regression and supervised Partial Least Squares (PLS). Modern sufficient dimension reduction techniques, like the ones we consider, constitute a form of supervised linear dimension reduction which outperform PCA and PLS without the underlying model assumptions. In practice, we often deal with situations where the same variables are being measured on objects from different groups, and we would like to know how similar the groups are with respect to some set of overall features. Common PCA and partial dimension reduction methods are extant methods for multiple groups. Note however that common PCA is unsupervised and doesn't take into account the information in Y and partial dimension reduction ignores the population-specific effects. Most importantly, these methods can not tell us if the same set of directions serve for all populations. To determine these common directions, we first propose a link-free procedure for testing whether two multi-index models share identical indices via the sufficient dimension reduction approach. Then a general method is introduced for two or more models based on modified partial dimension reduction. We present our test statistics, with associated asymptotic distributions and simulation studies. Applications to the well-known AIS data and the beta-carotene data are also used to demonstrate our methods. Furthermore, simulation studies are used to compare the various methods we consider"--Abstract, page iv.
Missouri University of Science and Technology
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