Quadratic functional estimation in inverse problems

In this paper, we consider a Gaussian sequence of independent observations having a polynomially increasing variance. This model describes a large panel of inverse problems, such as the deconvolution of blurred images or the recovering of the fractional derivative of a signal. We estimate the sum of squares of the means of our observations. This quadratic functional has practical meanings, e.g. the energy of a signal, and it is often used for goodness-of-fit testing.We compute Pinsker estimators when the underlying signal has both a finite and infinite amount of smoothness. When the signal is sufficiently smoother than the difficulty of the inverse problem, we attain the parametric rate and the efficiency constant associated with it. Moreover, we give upper bounds of the second order term in the risk. Otherwise, when the parametric rate cannot be attained, we compute non parametric upper bounds of the risk.