Pith Number

pith:MJMGSNKT

pith:2026:MJMGSNKTQV6IVQ4NOGFUKSXRJU

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A Grid-Rate Condition for Valid Uniform Inference

Emmanuel Selorm Tsyawo

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.

arxiv:2605.12284 v2 · 2026-05-12 · econ.EM

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-08T01:04:07.338926Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6258693553857c8ac38d718b454af14d035410128257706f2b9b5dd9cba75b0b

Aliases

arxiv: 2605.12284 · arxiv_version: 2605.12284v2 · doi: 10.48550/arxiv.2605.12284 · pith_short_12: MJMGSNKTQV6I · pith_short_16: MJMGSNKTQV6IVQ4N · pith_short_8: MJMGSNKT

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MJMGSNKTQV6IVQ4NOGFUKSXRJU \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 6258693553857c8ac38d718b454af14d035410128257706f2b9b5dd9cba75b0b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d19ca6d14f6c44c4e6f7cf9cdd057f3cd66cf30c206b35716b6505cd34e519f6",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "econ.EM",
    "submitted_at": "2026-05-12T15:42:02Z",
    "title_canon_sha256": "4a06c9e3746b02702516f6e1771d8cbf60569b6ab37130a807f10be34c161ad7"
  },
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  "source": {
    "id": "2605.12284",
    "kind": "arxiv",
    "version": 2
  }
}