{
 "bundle_of": "C10",
 "schema_version": "0.41",
 "paper": {
  "id": "entrylib:2026.000003",
  "version": 2,
  "title": "The observed-data likelihood of ADAM and iETS with unobserved periods",
  "licence": "CC-BY-4.0",
  "status": "corrected",
  "date": "2026-11-11",
  "status_notice": {
   "date": "2026-11-11",
   "reason_code": "correction",
   "statement": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed.",
   "issued_by": "Ivan Svetunkov (ORCID 0000-0001-7826-0281)"
  }
 },
 "claims": [
  {
   "id": "C10",
   "statement": "For a model with an occurrence part, occurrence probability $p_t$ and positive sizes, a prediction interval $[0,u]$ of the demand with coverage $c>1-p_t$ that counts the zeros as covered has as upper bound $u$ the quantile at level $(c-(1-p_t))/p_t$ of the distribution of the demand size, whose mean, the conditional mean of the size, is the size forecast $\\hat\\mu_t$, not the demand forecast $p_t\\hat\\mu_t$.",
   "about": {
    "subject": "a prediction interval of the demand that counts zeros as covered",
    "relation": "is an interval of the demand sizes centred on",
    "object": "the size forecast",
    "comparator": null,
    "conditions": [
     "models with an occurrence part"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "no_prior_found",
   "scope": {
    "population": "models with an occurrence part (iETS, ADAM)",
    "conditions": [
     "prediction interval of the demand of the form $[0,u]$ that counts zeros as covered",
     "coverage $c>1-p_t$",
     "Bernoulli occurrence independent of a positive continuous demand size"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E12"
    }
   ],
   "assumptions": [
    "C31",
    "C32"
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "relations": [
    {
     "type": "extends",
     "reference": "B4",
     "self": true,
     "note": "Svetunkov and Boylan forecast intermittent demand as the product of the occurrence probability and the forecast demand size."
    }
   ],
   "checks": [
    {
     "by": "P6",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "P(0<=y<=u)=(1-p)+pF_z(u) for positive sizes, so coverage c>1-p needs F_z(u)=(c-(1-p))/p, a quantile of the size distribution whose mean is mu_hat while E(y)=p mu_hat. Monte Carlo with p=0.4, c=0.9 gave coverage 0.8997 and size level 0.7500. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)"
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C31",
   "statement": "In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "theorem_condition",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C32",
   "statement": "The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "theorem_condition",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  }
 ],
 "evidence": [
  {
   "id": "E12",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E12.1",
     "statement": "$$P(0\\le y_t\\le u)=(1-p_t)+p_tF_z(u)$$ for positive sizes and occurrence independent of the size",
     "by": [
      "C31",
      "C32"
     ],
     "rule": "law of total probability"
    },
    {
     "id": "E12.2",
     "statement": "Setting it to $c$ with $c>1-p_t$ gives $F_z(u)=(c-(1-p_t))/p_t\\in(0,1]$, so $u$ is the quantile of the size at that level",
     "by": [
      "E12.1"
     ],
     "rule": "algebra"
    },
    {
     "id": "E12.3",
     "statement": "The size has mean $\\hat\\mu_t$, the demand $\\E(o_tz_t)=p_t\\hat\\mu_t$",
     "by": [
      "C31"
     ],
     "rule": "independence"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C31",
    "C32"
   ]
  }
 ],
 "artefacts": [
  {
   "id": "A1",
   "kind": "manuscript",
   "uri": "https://github.com/openforecast-org/smooth/blob/8a5fbbb334b0d6192a3c3e8a5773f512e548e891/python/docs/INTERMITTENT_LIKELIHOOD.tex",
   "version": "sha256:44eb855578262cd1668929bc6659cfcaaa4cd3b3e153541deffd91720908d1e3",
   "availability": {
    "level": "public"
   },
   "description": "LaTeX source of the draft 'Estimating exponential smoothing with unobserved periods: the observed-data likelihood of ADAM and iETS' (October 2026).",
   "licence": "LGPL-2.1-only"
  },
  {
   "id": "A10",
   "kind": "proof",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks.md",
   "version": "sha256:2ccd9457ef876f55c27b3b0004c25c5a991507f6667c59795e7515cc552fd7f1",
   "availability": {
    "level": "public"
   },
   "licence": "CC-BY-4.0",
   "description": "Report of the first round of the independent proof check by the contributor P6 (2026-11-11): for C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19, the re-derivation, the numerical spot checks, the outcome and the gaps found."
  },
  {
   "id": "A11",
   "kind": "proof",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks-2.md",
   "version": "sha256:1ba26c97d75792bd6d91d375288b585b375e459560b131ad4102b5762e7dbbf7",
   "availability": {
    "level": "public"
   },
   "licence": "CC-BY-4.0",
   "description": "Report of the second round of the independent proof check by P6 (2026-11-11), on C2, C5, C10, C14 and C19 as corrected after the first round, and on the smaller changes to C1, C6, C8 and C16."
  },
  {
   "id": "A14",
   "kind": "code",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk1.py",
   "version": "sha256:88de8878db174afda5c9a385d6d46bd0b00e744ce5fa8402814f47aa9d65db8f",
   "availability": {
    "level": "public"
   },
   "licence": "LGPL-2.1-only",
   "description": "Numerical spot checks of the first round of the proof check (Python with numpy); the report A10 says which claim each serves."
  },
  {
   "id": "A17",
   "kind": "code",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk4.py",
   "version": "sha256:aef765b48df441007f23a0a6bce88cc0b3b17256ea1a39f7fcdda49546002237",
   "availability": {
    "level": "public"
   },
   "licence": "LGPL-2.1-only",
   "description": "Numerical spot checks of the second round of the proof check (Python with numpy); the report A11 says which claim each serves."
  }
 ]
}