Simultaneous Inference for Non-Stationary Random Fields, with Application to Gridded Data Analysis

Abstract: Current statistics literature on statistical inference of random fields typically assumes that the fields are stationary or focuses on models of non-stationary Gaussian fields with parametric/semiparametric covariance families, which may not be sufficiently flexible to tackle complex modern-era random field data. This paper performs simultaneous nonparametric statistical inference for a general class of non-stationary and non-Gaussian random fields by modeling the fields as nonlinear systems with location-dependent transformations of an underlying `shift random field'. Asymptotic results, including concentration inequalities and Gaussian approximation theorems for high dimensional sparse linear forms of the random field, are derived. A computationally efficient locally weighted multiplier bootstrap algorithm is proposed and theoretically verified as a unified tool for the simultaneous inference of the aforementioned non-stationary non-Gaussian random field. Simulations and real-life data examples demonstrate good performances and broad applications of the proposed algorithm.