Abstract
We consider the nonparametric estimation problem of amultidimensional regression function. We propose to improve theoptimal rate of estimation from the minimax point of view. Inorder to avoid poor estimation quality or generally in models forwhich 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 minimaxhypothesis testing theory. In fact, this hybrid approach uses theresults of test theory to consider adaptive estimation. So, viathe concept of random normalizing factors introduced by Lepski,considering a ‘‘plausible’’ assumption that the regressionfunction has the single-index structure, we construct an estimatorthat can be adaptive and whose observation-dependent estimationrate is better than that obtained via the minimax approach, withprescribed confidence level \(\alpha_{n}\) . In addition, wedemonstrate the relevance of our results by applying them to realdata set.
