Sign‐flip inference for spatial regression with differential regularisation

.We address the problem of performing inference on the linear and nonlinear terms of a semiparametric spatial regression model with differential regularisation. For the linear term, we propose a new resampling procedure, based on (partial) sign‐flipping of an appropriate transformation of the residuals of the model. The proposed resampling scheme can mitigate the bias effect induced by the differential regularisation. We prove that the proposed test is asymptotically exact. Moreover, we show, by simulation studies, that it enjoys very good control of Type‐I error also in small sample scenarios, differently from parametric alternatives. Additionally, we show that the proposed test has higher power with respect than recently proposed nonparametric tests on the linear term of semiparametric regression models with differential regularisation. Concerning the nonlinear term, we develop three different inference approaches: a parametric one and two nonparametric alternatives. The nonparametric tests are based on a sign‐flip approach. One of these is proved to be asymptotically exact, while the other is proved to be exact also for finite samples. Simulation studies highlight the good control of Type‐I error of the nonparametric approaches with respect the parametric test, while retaining high power.