{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:50:26Z","timestamp":1791323426377,"version":"4.3.1"},"reference-count":50,"publisher":"Institute of Mathematical Statistics","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"The problem of estimating the slope parameter in regression between two spatial processes under confounding by an unmeasured spatial process has received widespread attention in the recent statistical literature. Yet, a fundamental question remains unresolved: when is this slope consistently estimable under spatial confounding, with existing insights being largely empirical or estimator-specific. In this manuscript, we characterize conditions for consistent estimability of the regression slope between Gaussian random fields (GRFs), the common stochastic model for spatial processes, under spatial confounding. Under fixed-domain (infill) asymptotics, we give sufficient conditions for consistent estimability in terms of the smoothness or local behavior of the exposure and confounder processes. When estimability holds, we provide consistent estimators of the slope using local differencing (taking discrete differences or Laplacians of the processes of suitable order). Using functional analysis results on Paley\u2013Wiener spaces, we then provide an easy-to-verify necessary condition for consistent estimability of the slope in terms of the relative spectral tail decays of the confounder and exposure. As a by-product, we establish a novel and general spectral condition on the equivalence of measures on the paths of multivariate GRFs with component fields of varying smoothnesses, a result of independent importance. Our estimability results or estimators do not rely on specific parametric models for the covariance functions. We show that for many covariance classes like the Mat\u00e9rn, power-exponential, generalized Cauchy and coregionalization families, the necessary and sufficient conditions become identical, except for a boundary point, thereby providing a sharp characterization of consistent estimability of the slope for these processes. The results are extended to multivariate slopes and to accommodate measurement error using local-averaging-and-differencing-based estimators. We show that differencing-based estimation remains consistent for popular classes of nonstationary Gaussian random fields, some non-Gaussian random fields and irregular designs. The finite sample behavior of the estimators is explored via numerical experiments.<\/jats:p>","DOI":"10.1214\/26-aos2650","type":"journal-article","created":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T12:51:15Z","timestamp":1790945475000},"page":"2406-2431","source":"Crossref","is-referenced-by-count":0,"title":["Infill consistent estimability of the regression slope between Gaussian random fields under spatial confounding"],"prefix":"10.1214","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5046-0289","authenticated-orcid":false,"given":"Abhirup","family":"Datta","sequence":"first","affiliation":[{"name":"Department of Biostatistics, Johns Hopkins University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael L.","family":"Stein","sequence":"additional","affiliation":[{"name":"Department of Statistics, Rutgers University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"ANDERES, E.<\/small> (2010). 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