Quadratic functional estimation in inverse problems

Abstract

We consider in this paper a Gaussian sequence model of observations $Y_i$, $i\geq 1$ having mean (or signal) $θ_i$ and variance $σ_i$ which is growing polynomially like $i^γ$, $γ>0$. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal $\sum_{i\geq 1}θ_i^2$ when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights $i^α$, $α>0$) and infinite smoothness (exponential weights $e^{βi^r}$, $β>0$, $0 γ+1/4$ or in the case of exponential weights), we obtain the parametric rate and the efficiency constant associated to it. Moreover, we give upper bounds of the second order term in the risk and conjecture that they are asymptotically sharp minimax. When the signal is finitely smooth with $α\leq γ+1/4$, we compute non parametric upper bounds of the risk of and we presume also that the constant is asymptotically sharp.