CLAIM C5 of entrylib:2026.000006 Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
licence CC-BY-4.0; contributions C5

DEFINITIONS
D1 [method] ETS(M,*,*): 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]
D4 [concept] Log-Normal error distribution: 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 $V(1+\epsilon_t) = e^{\sigma^2} - 1$.
D5 [concept] Gamma error distribution: 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 $V(1+\epsilon_t) = \sigma^2$.
D6 [concept] Inverse Gaussian error distribution: 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 $V(1+\epsilon_t) = \sigma^2$.
D7 [concept] Normal error distribution: Classical Gaussian distribution on $(-\infty, \infty)$ applied to $1+\epsilon_t$ with mean $\mu = 1$ and variance $\sigma^2$. [source B3]

CLAIMS
C5 [formal; principal; extension; support: formal_proof E3; assumes C12; verification unchecked; verifiability open; planning exploratory]
  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.
  scope: conditions: $ETS(M,*,*)$ models estimated via maximum likelihood on strictly positive observations
  extends isbn 978-3-540-71918-2, at p. 68: Generalises the Normal multiplicative ETS log-likelihood to Inverse Gaussian, Log-Normal, and Gamma distributions.
C12 [formal; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  The multiplicative error term $(1+\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\epsilon_t) = 1$.

EVIDENCE
E3 [formal; foundation mathematical_statistics] 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.
  assumes C12
  E3.1 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 S, e, c, t, i, o, n,  , 4, ., 1, ,,  , T, a, b, l, e,  , 2; moment_constraint)
  E3.2 Substitute parameter restrictions into probability density functions and Jacobian terms to obtain concentrated log-likelihood functions. (by S, e, c, t, i, o, n,  , 4, ., 1, ,,  , T, a, b, l, e,  , 3; likelihood_derivation)
  E3.3 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 S, e, c, t, i, o, n,  , 4, ., 1, ,,  , p, p, .,  , 9, -, 1, 0,  , a, n, d,  , e, n, t, r, y, l, i, b, :, 2, 0, 2, 6, ., 0, 0, 0, 0, 0, 3, @, v, 4, #, C, 7; maximum_likelihood_estimation)

REFERENCES
B3 978-3-540-71918-2 (isbn)
B7 10.1201/9781003452652 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C5, in: Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1#C5. https://entrylib.org/p/2026.000006/C5
The claimset it is from: Svetunkov, I. (guarantor) (2026). Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1. https://entrylib.org/p/2026.000006@v1