Minimax estimation of linear functionals under squared error loss

Under a very general setting, we consider the problem of estimating a linear functional of an unknown vector in a Hilbert space from indirect data contaminated by noise. We then discuss two situations in detail: estimating the signal function in the fractional Brownian motion model and the regression model with correlated errors. In the fractional Brownian motion model, we observe the process which is the sum of a fractional Brownian motion with Hurst index between ( 1 2 , 1 ) and a drift function that is determined by the signal function. In the regression model with correlated errors, we assume that the errors have long memory. For both estimation problems, we obtain the asymptotic rate for the minimax affine risks over certain types of parameter spaces. In each case, we also show that the minimax affine risk is bounded by 1.25 times the minimax risk.