L1-optimal estimates for a regression type function in rd

Abstract

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, θ(xi)), i = 1, ..., n. Assuming that θ lives in a sup-norm compact space Θq,d of real valued functions, an optimal L1-consistent estimator θ ̇n of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θq,d.