Detecting relevant changes in the spatiotemporal mean function

For a spatiotemporal process $\left\{X_j\right(s,t)\mid s\in S,t\in T\}{}_{j=1,\dots ,n}$, where $S$ denotes the set of spatial locations and $T$ the time domain, we consider the problem of testing for a change in the sequence of mean functions ${\left\{{\mu }_j\right(s,t)\mid s\in S,t\in T\}}_{j=1,\dots ,n}$. In contrast to most of the literature, we are not interested in arbitrarily small changes but only in changes with a norm exceeding a given threshold. Asymptotically distribution free tests are proposed, which do not require the estimation of the long-run spatiotemporal covariance structure. In particular, we consider a fully functional approach and a test based on the cumulative sum paradigm, investigate the large sample properties of the corresponding test statistics and study their finite sample properties by means of simulation study.