Efficiency in multivariate functional nonparametric models with autoregressive errors

In this paper, we introduce a new procedure for the estimation in the nonlinear functional regression model where the explanatory variable takes values in an abstract function space and the residual process is autocorrelated. Moreover, we consider the case where the response variable takes its values in RdRd. The procedure consists in a pre-whitening transformation of the dependent variable based on the estimated autocorrelation. We establish both consistency and asymptotic normality of the regression function estimate. For kernel methods encountered in the literature, the correlation structure is commonly ignored (the so-called “working independence estimator”); we show here that there is a strong benefit in taking into account the autocorrelation in the error process. We also find that the improvement in efficiency can be large in our functional setting, up to 25% in the presence of high autocorrelation levels. We observe that the additional step of iterating the fitting process actually deteriorates the estimation. We illustrate the skills of the methods on simulations as well as on application on ozone levels over the US.