Local adaptivity to inhomogeneous smoothness. 1. Resolution level
- Lepskii, Oleg V.
- Spokoiny, Vladimir
2010 Mathematics Subject Classification
- 62G07 62G20
The problem of nonparametric estimation of functions of inhomogeneous smoothness is considered. The goal is to define the notion of local smoothness of a function ƒ(·), to evaluate the optimal rate of convergence of estimators (depending on this local smoothness) and to construct an asymptotically efficient locally adaptive estimator. We treat local (or δ - local) smoothness properties of a function ƒ(·) at a point t as the corresponding characteristics of this function on the interval [t-δ,t+δ]. The value δ measures the ``locality'' of our procedure. The smaller this value is taken the more precise is our resolution analysis. But this value can not be taken arbitrary small since we should ce able to restore local smoothness properties of a function from the noisy data. The main result of the paper describes just the maximal rate of convergence of this parameter δ to zero as the noise level ε goes to zero. We call this value the resolution level. The value of this level strongly depends on the upper considered smoothness β* what we wish to attain. If κ*ε is the bandwidth corresponding to this smoothness β* then the resolution level δ*ε can not be chosen less (in order) than κ*ε. In particular, this yields that it is impossible to improve at the same time the accuracy of our procedure (which is measured by the upper smoothness β*) and its local adaptive properties. If we improve the accuracy of estimation at subintervals where a function is of high smoothness then we will have a low accuracy in a larger vicinity near a point with small smoothness. The main results claim that if the parameter of locality δ is taken less (in order) than the resolution level, then the corresponding risk is (asymptotically) infinite. After that we construct estimators with a finite asymptotic risk for the case of δ coinciding with the resolution level.
- Math. Methods Statistics, 4 (1995), pp. 239 -- 258