L1-optimal estimates for a regression type function in Rd

Let X1, X2, ..., Xn be random vectors that take values in a compact set in Rd, d >= 1. Let Y1, Y2, ..., Yn be random variables ("the responses") which conditionally on X1 = x1, ..., Xn = xn are independent with densities f(y xi, [theta](xi)), i = 1, ..., n. Assuming that [theta] lives in a sup-norm compact space [Theta]q,d of real valued functions, an optimal L1-consistent estimator of [theta] is constructed via empirical measures. The rate of convergence of the estimator to the true parameter [theta] depends on Kolmogorov's entropy of [Theta]q,d.