Testing the goodness-of-fit of a functional autoregressive model

The proposed Goodness–of–Fit (GoF) test for checking the linear autocorrelation model in a functional time series is based on an empirical process, whose residual marks and covariate index set are in a separable Hilbert space $\mathbb{H}$. A functional central limit theorem is derived providing the convergence of the empirical process to a time-changed Wiener process evaluated in a separable Hilbert space $\mathbb{H}$, with subordinator given by the marginal probability of the involved strictly stationary Autoregressive Hilbertian process (AR$\mathbb{H}$(1) process). The large sample behavior of the test statistics is obtained under simple and composite null hypotheses. The consistency of the test is addressed under simple null hypothesis. The finite-sample performance of the testing procedure, under different families of alternatives, and random projection schemes, is illustrated in the Appendix.