{
  "schema_version": "2.0",
  "seed_id": "solveall_adaptive_minimax_nonparametric_testing",
  "slug": "adaptive-minimax-nonparametric-testing",
  "split": "train",
  "title": "Adaptive Minimax Nonparametric Hypothesis Testing",
  "url": "https://solveall.org/problem/adaptive-minimax-nonparametric-testing",
  "categories": [
    "Mathematical Statistics"
  ],
  "statement": "Fix a dimension $d \\geq 1$, constants $L>0$, $\\alpha,\\beta \\in (0,1)$, and a known baseline function $f_0 \\in L_2([0,1]^d)$. In the periodic Gaussian white-noise model on $\\mathbb{T}^d=[0,1]^d$, $$\ndY(x)=f(x)\\,dx+n^{-1/2}dW(x), \\qquad x\\in \\mathbb{T}^d,\n$$ let $$\n\\mathcal W_2^s(L)=\\left\\{f\\in L_2(\\mathbb{T}^d):\\sum_{k\\in\\mathbb{Z}^d}(1+\\|k\\|_2^2)^s |\\theta_k(f)|^2\\le L^2\\right\\}, \\qquad s>0,\n$$ and test $$\nH_0:f=f_0\n\\quad\\text{vs}\\quad\nH_1(s,\\rho): f\\in \\mathcal W_2^s(L),\\ \\|f-f_0\\|_{L_2}\\ge \\rho.\n$$ For fixed $s$, the non-adaptive minimax separation radius is known to satisfy $$\n\\rho_n^*(s)\\asymp n^{-2s/(4s+d)}.\n$$\n\nClassical part (substantially understood): for several compact smoothness-range formulations (typically $s\\in[s_-,s_+]$ with $0<s_-<s_+<\\infty$ in Gaussian sequence/white-noise settings), exact adaptation ($c_n\\equiv 1$) is impossible and the optimal adaptive loss is of log-log type; in Spokoiny's normalization this appears as a factor $$\nt_\\varepsilon=(\\log\\log \\varepsilon^{-2})^{1/4},\n$$ with $\\varepsilon=n^{-1/2}$ (equivalently, a $(\\log\\log n)^{1/4}$-type factor in that parametrization).\n\nOpen problem: determine the sharp adaptive minimax rate outside those settled classical compact-range cases, especially for genuinely noncompact or otherwise broader regimes (for example $\\mathcal S=(0,\\infty)$, higher-dimensional/anisotropic families, or other model variations), and identify the minimal penalty $c_n(s,d,\\mathcal S)$ such that one test sequence controls type I/II errors uniformly over all $s\\in\\mathcal S$.",
  "unsolved_target": "Determine the sharp adaptive minimax rate outside those settled classical compact-range cases, especially for genuinely noncompact or otherwise broader regimes (for example $\\mathcal S=(0,\\infty)$, higher-dimensional/anisotropic families, or other model variations), and identify the minimal penalty $c_n(s,d,\\mathcal S)$ such that one test sequence controls type I/II errors uniformly over all $s\\in\\mathcal S$.",
  "sources": [
    {
      "source_id": "adaptive-minimax-inverse-2020",
      "title": "Adaptive minimax testing in inverse Gaussian sequence space models",
      "authors": [
        "Sandra Schluttenhofer",
        "Jan Johannes"
      ],
      "year": 2020,
      "venue": "arXiv preprint",
      "doi": null,
      "arxiv_id": "2002.07623",
      "url": "https://arxiv.org/abs/2002.07623",
      "version": "arXiv record dated February 18, 2020",
      "source_status": "verified",
      "source_note": "Primary arXiv record inspected; relevant as a model-variation result on simultaneous adaptation and unavoidable logarithmic deterioration.",
      "verification_notes": "Source identity and abstract verified. Registry authors are incorrect and must be replaced. The abstract explicitly states an additional log-factor for adaptive radii and says it is unavoidable."
    },
    {
      "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": "v1",
      "source_status": "verified",
      "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": "ArXiv primary record verifies title, authors, model, ellipsoid/tensor-product Fourier formulation, rate and sharp asymptotics, and the statement that the derived tests are inherently non-adaptive."
    },
    {
      "source_id": "ingster-suslina-2003",
      "title": "Nonparametric Goodness-of-Fit Testing Under Gaussian Models",
      "authors": [
        "Yuri Ingster",
        "Irina Suslina"
      ],
      "year": 2003,
      "venue": "Springer Series in Statistics (book)",
      "doi": "10.1007/978-0-387-21580-8",
      "arxiv_id": null,
      "url": null,
      "version": null,
      "source_status": "identified",
      "source_note": null,
      "verification_notes": null
    },
    {
      "source_id": "spokoiny-1996",
      "title": "Adaptive Hypothesis Testing Using Wavelets",
      "authors": [
        "Vladimir G. Spokoiny"
      ],
      "year": 1996,
      "venue": "The Annals of Statistics",
      "doi": "10.1214/aos/1032181163",
      "arxiv_id": null,
      "url": "https://www.wias-berlin.de/people/spokoiny/publications/6_Spokoiny_a1_96/1032181163.pdf",
      "version": null,
      "source_status": "verified",
      "source_note": "Section 2.3 (Adaptive testing), especially Theorems 2.2-2.3; the adaptation factor is stated as $t_\\varepsilon=(\\ln\\ln\\varepsilon^{-2})^{1/4}$ (not $\\varepsilon^{-1}$), pp. 2481-2482.",
      "verification_notes": "Primary published PDF verified. Section 2.3, Theorems 2.2–2.4, and equation (2.4) appear at the cited pages 2481–2482. Corrects the registry DOI."
    }
  ],
  "evidence_cards": [
    {
      "evidence_id": "inverse-model-adaptation-penalty",
      "reference_statement": "In the inverse Gaussian sequence model with noisy operator observations, Schluttenhofer and Johannes state that Bonferroni adaptation with respect to alternative regularity deteriorates testing radii by an additional log-factor, and that this deterioration is unavoidable under their assumptions.",
      "scope_conditions": [
        "Correct the authors to Sandra Schluttenhofer and Jan Johannes.",
        "Do not identify the abstract’s 'log-factor' with a universal exact exponent; its form depends on the model and assumptions.",
        "The result concerns the inverse model with noisy operator observations, not direct Gaussian white noise."
      ],
      "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": "machine_verified",
      "verification_notes": "Verified at the abstract level; theorem-level exact factor remains outside the supplied passage."
    },
    {
      "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": [
        "This is a multivariate random-design Gaussian regression benchmark, not the direct periodic Gaussian white-noise model.",
        "The source only briefly discusses adaptivity as an extension and does not establish the sharp adaptive penalty claimed in the SolveAll target."
      ],
      "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": "machine_verified",
      "verification_notes": "Fully verified from the arXiv primary record."
    },
    {
      "evidence_id": "spokoiny-compact-impossibility-repaired",
      "reference_statement": "In Spokoiny’s one-dimensional signal-plus-white-noise testing problem over Besov balls, adaptation at the fixed-parameter minimax rate is impossible for every nontrivial parameter set T: for every constant c>0 and every test, the null error plus the worst-case type-II error at separation cD_s(epsilon) is asymptotically at least 1.",
      "scope_conditions": [
        "The model is dX(t)=f(t)dt+epsilon dW(t), 0<=t<=1, with null hypothesis f=0.",
        "The alternatives are Besov-ball alternatives F_s(D) from Spokoiny’s Section 2, with the smoothness parameter tuple s=(s,p,q,M).",
        "T is nontrivial in the paper’s definition: there exist p,q,M and s_< < s_> such that (s,p,q,M) belongs to T for every s in [s_<,s_>].",
        "D_s(epsilon) is the fixed-parameter minimax testing rate defined in the paper, and beta_s is the supremum type-II error over the corresponding alternative.",
        "The conclusion concerns asymptotic total error, not a finite-sample exact-error statement."
      ],
      "passages": [
        {
          "source_id": "spokoiny-1996",
          "locator": "Published Annals of Statistics version, Section 2.3, definition of nontrivial T and Theorem 2.2, p. 2481",
          "text": "We say that a set T is nontrivial if there are such p, q, M and s' < s'' that (s, p, q, M) ∈ T, ∀ s ∈ [s', s'']. The first result shows that adaptive testing (without loss of power) is impossible for any nontrivial set T. THEOREM 2.2. Let T be nontrivial. Then for any c > 0 and any test φ, P0(φ = 1) + sup_{s∈T} β_s(φ, cD_s(ε)) ≥ 1 − o(1).",
          "content_sha256": "c710b758a5b8dcd72ca2fdb680a19816c2baeab4c1fd557f317a0750e90d1f71"
        }
      ],
      "verification_status": "machine_verified",
      "verification_notes": "Verified against the primary published PDF. The wording 'asymptotically at least 1' is appropriately interpreted as the displayed 1−o(1) lower bound."
    },
    {
      "evidence_id": "spokoiny-compact-upper-bound-repaired",
      "reference_statement": "For the bounded parameter family T specified in Spokoiny’s Theorem 2.4, there exists a single test whose null error and worst-case type-II error both tend to zero when the separation is a constant multiple of the fixed-parameter rate D_s(epsilon) times t_epsilon=(ln ln epsilon^{-2})^{1/4}.",
      "scope_conditions": [
        "The model is Spokoiny’s one-dimensional signal-plus-white-noise model with Besov alternatives.",
        "The parameter family is T={s=(s,p,q,M): s<=s̄, 1<=p<=p̄, M_lower<=M<=M_upper, sp>1/4}, with prescribed positive s̄,p̄ and M_lower<=M_upper, exactly as displayed in Theorem 2.4.",
        "The constant c depends on the displayed parameter bounds.",
        "The result is asymptotic: P0(φε=1)=o(1) and sup_{s∈T}β_s(φε,cD_s(ε)tε)=o(1).",
        "This theorem is an upper bound for the specified bounded Besov family; it does not cover the noncompact range S=(0,infinity) or the user’s d-dimensional periodic Sobolev family."
      ],
      "passages": [
        {
          "source_id": "spokoiny-1996",
          "locator": "Published Annals of Statistics version, 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 = (s, p, q, M): s ≤ s̄, 1 ≤ p ≤ p̄, M_lower ≤ M ≤ M_upper, sp > 1/4} with some prescribed positive s̄, p̄, M_lower ≤ M_upper. Then there exist a constant c = c(s̄, p̄, M_lower, M_upper) and a test φε such that P0(φε = 1) = o(1), sup_{s∈T} β_s(φε, cD_s(ε)tε) = o(1).",
          "content_sha256": "cf105f1826e9699aa6ed694913e100e70e86323ce3f5fd986aa3bbbebadd6652"
        }
      ],
      "verification_status": "machine_verified",
      "verification_notes": "Verified as an upper bound for the specified bounded Besov family only. The card correctly limits the theorem and explicitly excludes the noncompact and user-mentioned higher-dimensional settings."
    },
    {
      "evidence_id": "spokoiny-loglog-lower-bound-repaired",
      "reference_statement": "For every nontrivial Besov-parameter set T in Spokoiny’s framework, the fourth-root log-log factor t_epsilon=(ln ln epsilon^{-2})^{1/4} is a sharp lower-bound scale for adaptive deterioration: replacing it by any t'_epsilon with t'_epsilon/t_epsilon=o(1) leaves every test asymptotically powerless in the stated total-error sense.",
      "scope_conditions": [
        "The model and alternatives are exactly those of Spokoiny’s one-dimensional signal-plus-white-noise Besov problem.",
        "T is nontrivial as defined in Section 2.3.",
        "epsilon tends to zero.",
        "The candidate factor t'_epsilon satisfies t'_epsilon/t_epsilon=o(1).",
        "The conclusion is the theorem’s asymptotic lower bound for null error plus worst-case type-II error."
      ],
      "passages": [
        {
          "source_id": "spokoiny-1996",
          "locator": "Published Annals of Statistics version, Section 2.3, equation (2.4), p. 2482",
          "text": "tε = (ln ln ε−2)1/4.",
          "content_sha256": "ac8a354d6137b9fe87a0da085feb97e01a6454ca5435a8ad9f1c572d0fa8bb21"
        },
        {
          "source_id": "spokoiny-1996",
          "locator": "Published Annals of Statistics version, Section 2.3, Theorem 2.3, p. 2482",
          "text": "THEOREM 2.3. Let tε be as above. 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 φε, P0(φε = 1) + sup_{s∈T} βs(φε, cDs(ε)t′ε) ≥ 1 − o(1).",
          "content_sha256": "cf58ac9f7dff501bbb3202a379243989af6eaf37211b2ee5f71a63e71ee8e054"
        }
      ],
      "verification_status": "machine_verified",
      "verification_notes": "The cited theorem supports the sharp lower-bound scale in the stated Spokoiny framework. 'Asymptotically powerless' is a faithful paraphrase of the displayed total-error lower bound, not an additional finite-sample claim."
    }
  ],
  "priors": [],
  "generation_metadata": {
    "source_dataset": "data/solveall_dataset/problems.jsonl",
    "source_sha256": "8c495a0059526e41559ffa4abb93bb84ba4f48eb244ee7d4c5ae9ffc2792243f",
    "split_seed": 20260721,
    "evidence_research_model": "gpt-5.6-luna",
    "evidence_verifier_model": "gpt-5.6-luna",
    "candidate_evidence_cards": 5,
    "verified_evidence_cards": 5,
    "evidence_repair_model": "gpt-5.6-luna",
    "repaired_verified_evidence_cards": 3
  }
}