Abstract
We want to recover the regression function in the single-index model. Using an aggregation algorithm with local polynomial estimators, we answer in particular to the second part of Question 2 from Stone (1982) on the optimal convergence rate. The procedure constructed here has strong adaptation properties: it adapts both to the smoothness of the link function and to the unknown index. Moreover, the procedure locally adapts to the distribution of the design. We propose new upper bounds for the local polynomial estimator (which are results of independent interest) that allows a fairly general design. The behavior of this algorithm is studied through numerical simulations. In particular, we show empirically that it improves strongly over empirical risk minimization.
Highlights
The single-index model is standard in statistical literature
It is widely used in several fields, since it provides a simple trade-off between purely nonparametric and purely parametric approaches. It is well-known that it allows to deal with the so-called “curse of dimensionality” phenomenon. This phenomenon is explained by the fact that the minimax rate linked to this model is the same as in the univariate model
This paper provides new minimax results about the single-index model, which provides an answer, in particual, to the latter question
Summary
The single-index model is standard in statistical literature. It is widely used in several fields, since it provides a simple trade-off between purely nonparametric and purely parametric approaches. Even for small values of d (larger than 2), the dimension has a strong impact on the quality of estimation when no prior assumption on the structure of the multivariate regression function is made In this sense, the single-index model provides a simple way to reduce the dimension of the problem. It is assumed below that f belongs to some family of Holder balls, that is, we do not suppose its smoothness to be known Statistical literature on this model is wide. To prove the upper bound, we use an estimator which adapts both to the index parameter and to the smoothness of the link function This result is stated under fairly general assumptions on the design, which include any “non-pathological” law for PX. This estimator has a nice “design-adaptation” property, since it does not depend within its construction on PX
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