Minimax Risk with Random Normalizing Factors in the Single-Index Model

We consider the nonparametric estimation problem of a multidimensional regression function. We propose to improve the optimal rate of estimation from the minimax point of view. In order to avoid poor estimation quality or generally in models for which the minimax approach is unsatisfactory, Lepski [1] introduced the concept of random normalizing factors in $$1999$$ . This concept is a combination of adaptive estimation and minimax hypothesis testing theory. In fact, this hybrid approach uses the results of test theory to consider adaptive estimation. So, via the concept of random normalizing factors introduced by Lepski, considering a ‘‘plausible’’ assumption that the regression function has the single-index structure, we construct an estimator that can be adaptive and whose observation-dependent estimation rate is better than that obtained via the minimax approach, with prescribed confidence level $$\alpha_{n}$$ . In addition, we demonstrate the relevance of our results by applying them to real data set.