On convergence rates of suprema in the presence of non-negligible trends
- Konakov, Valentin
2010 Mathematics Subject Classification
- 62G07 62G20 62M40
- Kernel estimation, smoothing parameter, Gaussian fields, Laplace type integral, Leray-Gel'fand differential form
We investigate the convergence rates for the maximal deviation distribution of kernel estimates from the true density. The convergence rates for related Gaussian fields are also investigated. We consider the optimal choice of the smoothing parameter in the sense of Konakov and Piterbarg (1994) and in doing so we take into account a non-negligible trend. It is shown that the convergence rates depend on the asymptotic behaviour of the Laplace type integrals over a small neighbourhood of the manifold of points at which the trend attains its maximal value. Using integration over the level sets (Leray-Gel'fand differential forms) it is proved that the convergence rates are tipically logarithmically slow, even if the rates are to be uniform over as few as three points. Some improved approximations with power rates of convergence are also obtained.