Projection-based white noise and goodness-of-fit tests for functional time series

We develop two significance tests in the setting of functional time series. The null hypothesis of the first test is that the observed data are sampled from a general weak white noise sequence. The null hypothesis of the second test is that the observed data are sampled from a functional autoregressive model of order one (FAR(1)), which can be used as a goodness-of-fit test. Both tests are based on projections on functional principal components. Such projections are used in a great many functional data analysis (FDA) procedures, so our tests are practically relevant. We derive test statistics for each test that are quadratic forms of lagged autocovariance estimates constructed from principal component projections, and establish the requisite asymptotic theory. A simulation study shows that the tests have complimentary advantages against relevant benchmarks.