MINIMAX ESTIMATION OF LINEAR FUNCTIONALS IN THE CONVOLUTION MODEL

Consider the convolution model Yk = Xk + εk, k = 1, . . . , n, where the (Xk)’s and the (εk) ′s, are two independent sequences of independent and identically distributed random variables, the (Xk)’s with unknown density g and the (εk) ′s having the Gaussian density fε with zero mean and unit variance. In this model we aim at estimating, using the observations Y1, . . . , Yn, some linear functionals of the density g of the form Γf (y) = ∫ f(x)g(x)fε(y−x) dx, where f is a known function, either polynomial or trigonometric. We extend Taupin’s results [21] by giving lower bounds for pointwise minimax risk and upper and lower bounds for minimax Lp(R)-risk, when 2 ≤ p ≤ ∞.