Inference in nonlinear random fields and non-asymptotic rates for threshold variance estimators under sparse dependence

Inference based on the (functional) central limit theorem for nonlinear random fields is studied and generalized to the nonstationary case. For this purpose, nonparametric estimation of the variance of partial sums is studied in depth including a class of soft-thresholding estimators. Nonasymptotic convergence rates for all estimators are established. It is shown that threshold estimation is superior in terms of the convergence rate under a mild sparseness condition on the spatial covariance structure. The results also cover estimators calculated from residuals. Applications to hypothesis testing to detect effects such as tumors in CT images, regression models with external regressors, and sparse convolutional network layers are discussed.