Rate of Convergence in Density Estimation Using Neural Networks

Given N i.i.d. observations {X_i}^N_i=1 taking values in a compact subset of R^d, such that p^* denotes their common probability density function, we estimate p^* from an exponential family of densities based on single hidden layer sigmoidal networks using a certain minimum complexity density estimation scheme. Assuming that p^* possesses a certain exponential representation, we establish a rate of convergence, independent of the dimension d, for the expected Hellinger distance between the proposed minimum complexity density estimator and the true underlying density p^*.