{
 "bundle_of": "C5",
 "schema_version": "0.42",
 "paper": {
  "id": "entrylib:2026.000006",
  "version": 1,
  "title": "Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models",
  "licence": "CC-BY-4.0",
  "status": "active",
  "date": "2026-11-12"
 },
 "claims": [
  {
   "id": "C5",
   "statement": "Concentrated log-likelihood functions for [[D1:ETS(M,*,*)]] models under the unit-mean restriction $E(1+\\epsilon_t)=1$ take closed forms for [[D7:Normal error distribution]] $\\mathcal{N}(1,\\sigma^2)$, [[D6:Inverse Gaussian error distribution]] $\\mathcal{IG}(1,\\sigma^2)$, [[D4:Log-Normal error distribution]] $\\log\\mathcal{N}(-\\sigma^2/2,\\sigma^2)$, and [[D5:Gamma error distribution]] $\\Gamma(\\sigma^{-2},\\sigma^2)$ error distributions.",
   "about": {
    "subject": "D1",
    "relation": "admits closed-form concentrated log-likelihood functions across four error distributions",
    "object": null,
    "comparator": null,
    "conditions": [
     "unit expectation restriction $E(1 + \\epsilon_t) = 1$"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "extension",
   "planning": "exploratory",
   "scope": {
    "conditions": [
     "$ETS(M,*,*)$ models estimated via maximum likelihood on strictly positive observations"
    ]
   },
   "relations": [
    {
     "type": "extends",
     "target": "978-3-540-71918-2",
     "id_type": "isbn",
     "locator": "p. 68",
     "note": "Generalises the Normal multiplicative ETS log-likelihood to Inverse Gaussian, Log-Normal, and Gamma distributions."
    }
   ],
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E3"
    }
   ],
   "assumptions": [
    "C12"
   ],
   "verification": "unchecked",
   "provenance": "author_stated",
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C12",
   "statement": "The multiplicative error term $(1+\\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\\epsilon_t) = 1$.",
   "type": "formal",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "statistical",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  }
 ],
 "definitions": [
  {
   "id": "D1",
   "term": "ETS(M,*,*)",
   "kind": "method",
   "definition": "Pure multiplicative exponential smoothing state space model family written in logarithmic state space form as $$\\begin{aligned} \\log y_t &= w' \\log(v_{t-l}) + \\log(1 + \\epsilon_t) \\\\ \\log v_t &= F \\log v_{t-l} + \\log(1_k + g \\epsilon_t) \\end{aligned}$$ with conventional multi-step point forecast $\\hat{y}_{t+h} = \\exp\\left((w_1' F_1^{h-1} + w_m' F_m^{\\lceil h/m \\rceil - 1}) \\log v_t\\right)$.",
   "source": "B7"
  },
  {
   "id": "D4",
   "term": "Log-Normal error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with location $\\mu = -\\sigma^2/2$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = e^{\\sigma^2} - 1$."
  },
  {
   "id": "D5",
   "term": "Gamma error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with shape $\\xi = \\sigma^{-2}$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = \\sigma^2$."
  },
  {
   "id": "D6",
   "term": "Inverse Gaussian error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with location $\\mu = 1$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = \\sigma^2$."
  },
  {
   "id": "D7",
   "term": "Normal error distribution",
   "kind": "concept",
   "definition": "Classical Gaussian distribution on $(-\\infty, \\infty)$ applied to $1+\\epsilon_t$ with mean $\\mu = 1$ and variance $\\sigma^2$.",
   "source": "B3"
  }
 ],
 "evidence": [
  {
   "id": "E3",
   "profile": "formal",
   "description": "Analytical derivation of unit-mean parameter restrictions, concentrated log-likelihoods, and scale estimators for Normal, Inverse Gaussian, Log-Normal, and Gamma distributions in $ETS(M,*,*)$ models (Section 4.1, Tables 2-3), with author erratum referencing `entrylib:2026.000003@v4#C7` for the Log-Normal scale MLE.",
   "foundation": "mathematical_statistics",
   "assumptions": [
    "C12"
   ],
   "steps": [
    {
     "id": "E3.1",
     "statement": "Impose $E(1 + \\epsilon_t) = 1$ on each distribution: $\\mu = 1$ for Normal and Inverse Gaussian, $\\mu = -\\sigma^2 / 2$ for Log-Normal, and $\\xi = \\sigma^{-2}$ for Gamma.",
     "by": "Section 4.1, Table 2",
     "rule": "moment_constraint"
    },
    {
     "id": "E3.2",
     "statement": "Substitute parameter restrictions into probability density functions and Jacobian terms to obtain concentrated log-likelihood functions.",
     "by": "Section 4.1, Table 3",
     "rule": "likelihood_derivation"
    },
    {
     "id": "E3.3",
     "statement": "Differentiate log-likelihoods with respect to $\\sigma^2$ to obtain closed-form scale MLEs for Normal and Inverse Gaussian, method-of-moments estimator for Gamma, and the corrected Log-Normal scale MLE referenced in `entrylib:2026.000003@v4#C7`.",
     "by": "Section 4.1, pp. 9-10 and entrylib:2026.000003@v4#C7",
     "rule": "maximum_likelihood_estimation"
    }
   ],
   "availability": {
    "level": "public"
   }
  }
 ],
 "references": [
  {
   "id": "B3",
   "target": "978-3-540-71918-2",
   "id_type": "isbn"
  },
  {
   "id": "B7",
   "target": "10.1201/9781003452652",
   "id_type": "doi"
  }
 ]
}