Adaptive estimation of a quadratic functional by model selection

摘要: We consider the problem of estimating $\|s\|^2$ when $s$ belongs to some separable Hilbert space and one observes Gaussian process $Y(t) = \langles, t\rangle + \sigmaL(t)$, for all $t \epsilon \mathbb{H}$,where $L$ is isonormal process. This framework allows us in particular classical “Gaussian sequence model” which $\mathbb{H} l_2(\mathbb{N}*)$ $L(t) \sum_{\lambda\geq1}t_{\lambda}\varepsilon_{\lambda}$, where $(\varepsilon_{\lambda})_{\lambda\geq1}$ a i.i.d. standard normal variables. Our approach consists considering at most countable families finite-dimensional linear subspaces $\mathbb{H}$ (the models) then using model selection via conveniently penalized least squares criterion build new estimators $\|s\|^2$. prove general nonasymptotic risk bound show that such are adaptive on variety collections sets parameter $s$, depending family models from they built.In particular, context model, convenient choice defining over hyperrectangles, ellipsoids, $l_p$-bodies or Besov bodies.We take special care describe conditions under estimator efficient level noise $\sigma$ tends zero. construction an alternative by Efroimovich Low hyperrectangles provides results otherwise.