Self-normalized inference for stationarity of irregular spatial data

A self-normalized approach for testing the stationarity of a $d$-dimensional random field is considered in this paper. Because the discrete Fourier transforms (DFT) at fundamental frequencies of a second-order stationary random field are asymptotically uncorrelated (see Bandyopadhyay and Subba Rao, 2017), one can construct a stationarity test based on the sample covariance of the DFTs. Such a test is usually inferior because it involves an overestimated scale parameter that leads to low size and power. To circumvent this shortcoming, this paper proposes two self-normalized statistics based on extreme value and partial sum of the sample covariance of the DFTs. Under certain regularity conditions, it is shown that the proposed tests converge to functionals of Brownian motion. Simulations and a data analysis demonstrate the outstanding performance of the proposed tests.