  The presented methodology for testing the goodness-of-fit of an
Autoregressive Hilbertian model (ARH(1) model) provides an infinite-dimensional
formulation of the approach proposed in Koul and Stute (1999), based on
empirical process marked by residuals. Applying a central and functional
central limit result for Hilbert-valued martingale difference sequences, the
asymptotic behavior of the formulated H-valued empirical process, also indexed
by H, is obtained under the null hypothesis. The limiting process is H-valued
generalized (i.e., indexed by H) Wiener process, leading to an asymptotically
distribution free test. Consistency of the test is also proved. The case of
misspecified autocorrelation operator of the ARH(1) process is addressed. The
asymptotic equivalence in probability, uniformly in the norm of H, of the
empirical processes formulated under known and unknown autocorrelation operator
is obtained. Beyond the Euclidean setting, this approach allows to implement
goodness of fit testing in the context of manifold and spherical functional
autoregressive processes.
