L1-optimal estimates for a regression type function in rd
Journal
Journal of Multivariate Analysis
Date Issued
February 1992
Author(s)
DOI
10.1016/0047-259X(92)90023-9
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.
Subjects