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"schema_version": "2.0",
"artifact_type": "evidence_research_draft",
"seed_id": "solveall_adaptive_minimax_nonparametric_testing",
"problem_input_sha256": "2cc022e32d6fc798e72f6d12ed7ed976cb3131723a42b7322c66455d7565c39d",
"researcher_model": "gpt-5.6-luna",
"prompt_version": "solveall-evidence-research-v1",
"created_at": "2026-07-27T01:58:15.729606+00:00",
"research_notes": "Inspected the registered Spokoiny source via the author-hosted Annals of Statistics PDF and located Section 2.3, Theorems 2.2–2.4 (pp. 2481–2482). Also inspected the arXiv record for Ingster–Sapatinas (2009) and its full-text mathematical-statistics PDF, plus the arXiv record for adaptive minimax testing in inverse Gaussian sequence models (2020). The evidence supports the classical compact/nontrivial-range log-log penalty, gives a multivariate non-adaptive benchmark whose tests are explicitly described as non-adaptive, and documents a different model variation where adaptation incurs an unavoidable log-factor. I did not find an inspected primary source proving a sharp result for the genuinely noncompact range S=(0,∞) in the exact periodic white-noise Sobolev formulation; no claim of resolution for that regime is made.",
"new_sources": [
{
"source_id": "ingster-sapatinas-2009",
"title": "Minimax Goodness-of-Fit Testing in Multivariate Nonparametric Regression",
"authors": [
"Yuri I. Ingster",
"Theofanis Sapatinas"
],
"year": 2009,
"venue": "Mathematical Methods of Statistics",
"doi": "10.3103/S1066530709030041",
"arxiv_id": "0910.0936",
"url": "https://arxiv.org/abs/0910.0936",
"version": null,
"source_status": "identified",
"source_note": "Primary arXiv record and author-hosted/full-text journal PDF inspected; studies multivariate ellipsoid alternatives and explicitly characterizes the proposed tests as non-adaptive.",
"verification_notes": null
},
{
"source_id": "adaptive-minimax-inverse-2020",
"title": "Adaptive minimax testing in inverse Gaussian sequence space models",
"authors": [
"Mathilde Carpentier",
"T. Schluttenhofer"
],
"year": 2020,
"venue": "arXiv preprint",
"doi": null,
"arxiv_id": "2002.07623",
"url": "https://arxiv.org/abs/2002.07623",
"version": null,
"source_status": "identified",
"source_note": "Primary arXiv record inspected; relevant as a model-variation result on simultaneous adaptation and unavoidable logarithmic deterioration.",
"verification_notes": null
}
],
"evidence_cards": [
{
"evidence_id": "spokoiny-compact-impossibility",
"reference_statement": "In Spokoiny’s wavelet testing framework, for any nontrivial smoothness-parameter set T, no single test can retain the fixed-parameter minimax rate uniformly over T without a multiplicative loss.",
"scope_conditions": [
"The alternative is the wavelet/Besov-type formulation used in Spokoiny (1996), with unknown parameters s,p,q,M.",
"T must be nontrivial in the paper’s sense: it contains a continuum of s-values while the other parameters remain in a prescribed admissible range.",
"The conclusion is asymptotic and concerns the sum of type-I error and the supremum type-II error."
],
"passages": [
{
"source_id": "spokoiny-1996",
"locator": "Section 2.3, Theorem 2.2, p. 2481",
"text": "The first result shows that adaptive testing (without loss of power) is impossible for any nontrivial set T.",
"content_sha256": "1d8f219dcc83167907d2dfed7ea830c419bfa4ea3e80db4bb25f32d025b0d6eb"
},
{
"source_id": "spokoiny-1996",
"locator": "Section 2.3, Theorem 2.2, p. 2481",
"text": "THEOREM 2.2. Let T be nontrivial. Then for any c > 0 and any test φ, P0(φ = 1) + sup_{s∈T} βs(φ, c Ds(ε)) ≥ 1 − o(1).",
"content_sha256": "a71527b5c66c380b3220cd02dea7d5311f2c7bc1ba7e0566a4bd357e9d237744"
}
],
"verification_status": "pending",
"verification_notes": null
},
{
"evidence_id": "spokoiny-loglog-lower-bound",
"reference_statement": "For the same nontrivial parameter sets, Spokoiny identifies a fourth-root log-log adaptive factor as a necessary lower-bound penalty: any factor smaller than tε=(ln ln ε^{-2})^{1/4} in the stated sense cannot yield uniformly successful adaptation.",
"scope_conditions": [
"The noise level is ε→0 in Spokoiny’s notation.",
"The lower bound applies to nontrivial T as defined in Section 2.3.",
"The theorem assumes the candidate factor t′ε satisfies t′ε/tε=o(1).",
"The result is for the same wavelet/Besov-type testing problem and asymptotic error criterion as Theorem 2.2."
],
"passages": [
{
"source_id": "spokoiny-1996",
"locator": "Section 2.3, immediately before Theorem 2.3, p. 2481",
"text": "The next results show that for the problem under consideration the minimal adaptive factor is (ln ln ε−2)1/4.",
"content_sha256": "27858cf86397c4a4e39eb004f3984f733912b16cd8eb016ef47754ad819ed45d"
},
{
"source_id": "spokoiny-1996",
"locator": "Section 2.3, equation (2.4), p. 2482",
"text": "tε = (ln ln ε−2)1/4.",
"content_sha256": "ac8a354d6137b9fe87a0da085feb97e01a6454ca5435a8ad9f1c572d0fa8bb21"
},
{
"source_id": "spokoiny-1996",
"locator": "Section 2.3, Theorem 2.3, p. 2482",
"text": "If T is a nontrivial set and if t′ε is such that t′ε/tε = o(1), then for any c > 0 and any test φ, P0(φ = 1) + sup_{s∈T} βs(φ, c Ds(ε)t′ε) ≥ 1 − o(1).",
"content_sha256": "356aa57d581748e37c92a8d62d888b31b6690be23324f694b42c4b6d0787960f"
}
],
"verification_status": "pending",
"verification_notes": null
},
{
"evidence_id": "spokoiny-compact-upper-bound",
"reference_statement": "For a bounded, compactly parameterized range of smoothness and the other wavelet parameters, Spokoiny constructs a single adaptive test attaining the fixed-parameter rate multiplied by tε=(ln ln ε−2)^{1/4}, up to a constant.",
"scope_conditions": [
"T has the form s≤s̄, 1≤p≤p̄, M̲≤M≤M̄, with positive prescribed bounds and sp>1/4 as displayed in the theorem.",
"The conclusion is for the wavelet/Besov formulation and requires a constant c depending on the parameter bounds.",
"The type-I error tends to zero and the supremum type-II error over T tends to zero at separation c Ds(ε)tε."
],
"passages": [
{
"source_id": "spokoiny-1996",
"locator": "Section 2.3, Theorem 2.4, p. 2482",
"text": "THEOREM 2.4. Let tε be as above and let a set T be of the form T = {s,p,q,M : s ≤ s̄, 1 ≤ p ≤ p̄, M̲ ≤ M ≤ M̄, sp > 1/4} with some prescribed positive s̄, p̄, M̲ ≤ M̄. Then there exist a constant c = c(s̄,p̄,M̲,M̄) and a test φ such that P0(φ = 1) = o(1), sup_{s∈T} βs(φ, c Ds(ε)tε) = o(1).",
"content_sha256": "3429a3a456ba21aa3ca3f9e1b3f3551b30bdf70c651150ec079404518fb47126"
}
],
"verification_status": "pending",
"verification_notes": null
},
{
"evidence_id": "multivariate-nonadaptive-benchmark",
"reference_statement": "Ingster and Sapatinas develop rate and sharp-asymptotic minimax goodness-of-fit tests for multivariate ellipsoid alternatives, including multidimensional Sobolev and tensor-product Sobolev classes, but explicitly characterize the derived tests as non-adaptive.",
"scope_conditions": [
"The model is multivariate nonparametric regression with random uniform design on [0,1]^d, 1≤d≤∞, and Gaussian errors of known variance.",
"The alternative is an ellipsoid in L2([0,1]^d) with respect to a tensor-product Fourier basis, separated from a known f0 by rn.",
"This is not the periodic Gaussian white-noise model in the SolveAll statement; it is a related multivariate regression benchmark.",
"The source discusses adaptivity only as an extension, not as a theorem establishing a sharp adaptive penalty."
],
"passages": [
{
"source_id": "ingster-sapatinas-2009",
"locator": "Abstract, journal PDF p. 241 / arXiv abstract",
"text": "We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms",
"content_sha256": "72b88adc2075eabd7c5e94898607059497b15f05c18531f243116eb80fb8cd7f"
},
{
"source_id": "ingster-sapatinas-2009",
"locator": "Introduction, equations (1.2)–(1.3), journal PDF pp. 241–242",
"text": "Given a positive sequence rn → 0 as n → ∞ and a known function f0 ∈ L2(Δ) ... we propose, under general conditions, a unified framework for the goodness-of-fit testing problem for testing the null hypothesis H0 : f = f0 against the alternative H1 : f ∈ F, ||f − f0|| ≥ rn, where F is an ellipsoid in the Hilbert space L2(Δ) with respect to the tensor product Fourier basis",
"content_sha256": "dd86603f9442e91e022d963ff907db3ceb1024bddfce367ddf59f07519aa55ee"
}
],
"verification_status": "pending",
"verification_notes": null
},
{
"evidence_id": "inverse-model-adaptation-penalty",
"reference_statement": "In an inverse Gaussian sequence model with noisy operator observations, adaptive tests over ellipsoid regularity classes have radii deteriorated by an additional logarithmic factor, and the authors state that this deterioration is unavoidable under their assumptions.",
"scope_conditions": [
"The result concerns an inverse Gaussian sequence-space model with additional noisy observations of the operator, not direct Gaussian white noise.",
"It covers both signal detection and goodness-of-fit testing against a prescribed sequence.",
"The adaptation is with respect to regularity of the alternative; the exact logarithmic factor and its form depend on the model and assumptions in the paper.",
"The source states the unavoidable penalty at the abstract level; an exact theorem-level passage was not available in the inspected arXiv HTML record."
],
"passages": [
{
"source_id": "adaptive-minimax-inverse-2020",
"locator": "arXiv abstract, lines 3–6",
"text": "Furthermore, we apply a classical Bonferroni method for making both the indirect and the direct test adaptive with respect to the regularity of the alternative. The radii of the adaptive tests are deteriorated by an additional log-factor, which we show to be unavoidable.",
"content_sha256": "b9f8440f8072ab1cb1b4d2c31ef3c438b3626908a4d2a875c297c91c854b530a"
}
],
"verification_status": "pending",
"verification_notes": null
}
]
}
|