Nonparametric Estimation of a Linear Functional of the Regression Function for a Given Observation Design

In [Probl. Peredachi Inf., 1986, vol. 22, no. 3, pp. 43–61] we considered nonparametric estimation of a linear functional of the regression function for a variable observation design. In this paper, we consider the same problem for a given observation design $(t_1,\dots,t_n)$ such that $t_1,\dots,t_n$ are independent identically distributed random variables with a known distribution density. We obtain a`lower bound on estimation accuracy and, under relatively light assumptions on the set of allowed regression functions, construct an estimator that attains this bound.