Optimal rates and adaptation in the single-index model using aggregation

Details Optimal rates and adaptation in the single-index model using aggregation Optimal rates and adaptation in the single-index model using aggregation

2007-03-23

arxiv.org/abs/math/0703706v2

Electronic Journal of Statistics Volume 1, Issue 0 (2007) Pages 538-573

Mathematics

Statistics Theory

36 pages

Scientific paper

10.1214/07-EJS077

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.

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