2009/02/13 by Butucea, Cristina, Méziani, Katia · 1 citation
#62F12 #62G05 #62G10 #62G20 #FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.0902.2309
We consider in this paper a Gaussian sequence model of observations Yi, i≥ 1 having mean (or signal) θi and variance σi which is growing polynomially like iγ, γ>0. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal ∑i≥ 1θi2 when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights iα, α>0) and infinite smoothness (exponential weights eβir, β>0, 0γ+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≤ γ+1/4, we compute non parametric upper bounds of the risk of and we presume also that the constant is asymptotically sharp.