Parametric bootstrap for asymptotic test of contrast difference in neural networks

This work concernes the contrast difference test and its asymptotic properties for non linear auto-regressive models. Our approach is based on an application of the parametric bootstrap method. It is a re-sampling method based on the estimate parameters of the models. The resulting methodology is illustrated by simulations of multilayer perceptron models, and an asymptotic justification is given at the end. 1 Non Linear Auto-Regressive Models Let be two positive integers. A functional auto-regressive process on is a sequence of random vectors defined by: "!$#&%' (1) where %( is an i.i.d. noise with mean 0 and constant variance )$* , and where function is known. The parameter + belongs to a subset , of ( .0/21 3 ). Such a model is denoted below by 465 ! . Let 787 797 be the Euclidean norm on . We define the contrast process associated to the least squares by: