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Index models with unspecified ridge functions have received a considerable attention in the statistical literature for offering more flexibility than linear models and their ability of reducing dimension. We investigate the simple model where the covariate vector is projected along a single direction and the ridge function is assumed to be monotone. In this monotone single index model, we investigate the asymptotic performance of the Least square Estimator (LSE) of both the parametric and non-parametric components of the model. Under some regularity conditions, we show that the LSE of the true index and monotone ridge function converge at the typical cubic rate. Faster adaptive rates can be obtained in case the ridge functions is piece-wise constant with $m$ pieces. We describe a simple stochastic search algorithm to compute the estimator. We illustrate the theory through simulations that were performed under different scenarios.
This is a joint woork with Cecile Durot and Hanna K. Jankowski
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