{"paper":{"title":"A Grid-Rate Condition for Valid Uniform Inference","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For functions in a Donsker class, the grid-growth condition L_n = ω(r_n^{1/4}) suffices for valid uniform inference on twice continuously differentiable functions.","cross_cats":[],"primary_cat":"econ.EM","authors_text":"Emmanuel Selorm Tsyawo","submitted_at":"2026-05-12T15:42:02Z","abstract_excerpt":"Conducting uniform inference on a continuous functional F defined on a compact subset X of R^d involves specifying L_n^d nodes for estimation and the construction of confidence bands. While asymptotically valid inference requires L_n to increase with n, existing fixed-L rules of thumb and heuristic data-driven approaches lack formal justification. This paper shows that, for functions within a Donsker class, the simple grid-growth condition r_n^(1/4)/L_n -> 0, equivalently L_n grows faster than r_n^(1/4), is sufficient for valid inference on twice continuously differentiable functions whose est"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for functions within a Donsker class, the simple grid-growth condition L_n=ω(r_n^{1/4}) is sufficient for valid inference for twice continuously differentiable functions estimable at the r_n^{1/2} rate. This condition ensures that the approximation error is asymptotically negligible relative to the stochastic variation of the empirical process.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The approximation error from discretizing the domain onto L_n^d grid points is asymptotically negligible relative to the stochastic variation of the empirical process; this relies on the target function being twice continuously differentiable and belonging to a Donsker class.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"L_n = ω(r_n^{1/4}) ensures valid uniform inference for twice differentiable Donsker-class functions estimated at the r_n^{1/2} rate by making grid approximation error negligible relative to stochastic variation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For functions in a Donsker class, the grid-growth condition L_n = ω(r_n^{1/4}) suffices for valid uniform inference on twice continuously differentiable functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a1505d9e79d686e8679086c06dac779e5e026f2ccd33b68156ab10b25f93a5ec"},"source":{"id":"2605.12284","kind":"arxiv","version":2},"verdict":{"id":"5d770a41-a45e-4c5f-a21a-a2f829c18a5a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T02:36:17.868730Z","strongest_claim":"for functions within a Donsker class, the simple grid-growth condition L_n=ω(r_n^{1/4}) is sufficient for valid inference for twice continuously differentiable functions estimable at the r_n^{1/2} rate. This condition ensures that the approximation error is asymptotically negligible relative to the stochastic variation of the empirical process.","one_line_summary":"L_n = ω(r_n^{1/4}) ensures valid uniform inference for twice differentiable Donsker-class functions estimated at the r_n^{1/2} rate by making grid approximation error negligible relative to stochastic variation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The approximation error from discretizing the domain onto L_n^d grid points is asymptotically negligible relative to the stochastic variation of the empirical process; this relies on the target function being twice continuously differentiable and belonging to a Donsker class.","pith_extraction_headline":"For functions in a Donsker class, the grid-growth condition L_n = ω(r_n^{1/4}) suffices for valid uniform inference on twice continuously differentiable functions."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.12284/integrity.json","findings":[{"note":"Identifier '10.1007/978-0-387-76635-1' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. The cited work could not be located through any authoritative source.","detector":"doi_compliance","severity":"critical","ref_index":1,"audited_at":"2026-05-19T07:42:28.974789Z","detected_doi":"10.1007/978-0-387-76635-1","finding_type":"unresolvable_identifier","verdict_class":"cross_source","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-26T14:44:38.829150Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T14:01:25.414133Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T10:16:40.041282Z","status":"completed","version":"1.0.0","findings_count":1},{"name":"claim_evidence","ran_at":"2026-05-19T22:41:58.310054Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e9ba79e640747a3b07565a1fd4ba61e72df278f1483f09f3ebaf3d70b3165c21"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"618d2ef717696c656a0d8c8b24511f8dfe5be3d86a36e6ac4021fdf565d41e9d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}