Predict a new response from a covariate is a challenging task in regression, which raises new question since the era of high-dimensional data. In this paper, we are interested in the inverse regression method from a theoretical viewpoint. Theoretical results have already been derived for the well-known linear model, but recently, the curse of dimensionality has increased the interest of practitioners and theoreticians into generalization of those results for various estimators, calibrated for the high-dimension context. To deal with high-dimensional data, inverse regression is used in this paper. It is known to be a reliable and efficient approach when the number of features exceeds the number of observations. Indeed, under some conditions, dealing with the inverse regression problem associated to a forward regression problem drastically reduces the number of parameters to estimate and make the problem tractable. When both the responses and the covariates are multivariate, estimators constructed by the inverse regression are studied in this paper, the main result being explicit asymptotic prediction regions for the response. The performances of the proposed estimators and prediction regions are also analyzed through a simulation study and compared with usual estimators.
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