CLAIM C4 of entrylib:2026.000006 Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
licence CC-BY-4.0; contributions C4

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 $V(1+\epsilon_t) = \sigma^2$.
D10 [method] Akram's log-ETS model: Additive error ETS model $ETS(A,Ad,A)$ with Normal errors applied to the natural logarithms of strictly positive time series and exponentiated back to the original scale. [source B4]

CLAIMS
C4 [methodological; principal; extension; support: formal_proof E3, cited B4, at Section 2; assumes C12, C13; strength high (author); verification unchecked; verifiability open; planning exploratory]
  [[D4:Log-Normal error distribution]], [[D5:Gamma error distribution]], and [[D6:Inverse Gaussian error distribution]] distributions defined on $(0, \infty)$ can be applied directly to the multiplicative error term $(1+\epsilon_t)$ of conventional [[D1:ETS(M,*,*)]] models without transforming the model into [[D10:Akram's log-ETS model]], provided $E(1+\epsilon_t)=1$.
  scope: conditions: strictly positive time series modelled within the original $ETS(M,*,*)$ taxonomy
  extends doi 10.1111/j.1467-842X.2009.00555.x, at Section 2: Implements Log-Normal, Gamma, and Inverse Gaussian distributions directly within original $ETS(M,*,*)$ rather than sidestepping the taxonomy via logarithmic ETS.
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 $E(1+\epsilon_t) = 1$.
C13 [formal; supporting; assumption (methodological, tested); verifiability not_applicable]
  Smoothing parameters $\alpha, \beta, \gamma$ in pure multiplicative ETS models lie within $[0, 1]$ so that state update terms $(1 + g\epsilon_t)$ remain strictly positive whenever $1 + \epsilon_t > 0$.

EVIDENCE
E1 [formal; foundation classical_probability] Analytical derivation of recursive conditional expectations and point forecasts for pure multiplicative ETS models in logarithmic state space form (Section 2, Eqs. 2.1-2.11).
  assumes C12, C13
  E1.1 Reformulate pure multiplicative ETS in logarithmic state space form: $\log y_t = w' \log(v_{t-l}) + \log(1 + \epsilon_t)$ and $\log v_t = F \log v_{t-l} + \log(1_k + g \epsilon_t)$. (by S, e, c, t, i, o, n,  , 2, ,,  , E, q, .,  , (, 2, ., 1, ); logarithmic_transformation)
  E1.2 Expand $h$-steps-ahead recursion into non-seasonal and seasonal components and take conditional expectation $E(\log y_{t+h}|t)$. (by S, e, c, t, i, o, n,  , 2, ,,  , E, q, s, .,  , (, 2, ., 5, ), -, (, 2, ., 6, ); recursive_substitution)
  E1.3 Apply Jensen's inequality and non-negativity of $g$ to show point forecast $\hat{y}_{t+h} \le \tilde{y}_{t+h} \le \mu_{y,t+h}$, with equality only at $h=1$, $h \le m$ without trend, or for $ETS(M,N,N)$. (by S, e, c, t, i, o, n,  , 2, ,,  , E, q, s, .,  , (, 2, ., 8, ), -, (, 2, ., 1, 1, ); jensens_inequality)
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
B4 10.1111/j.1467-842X.2009.00555.x (doi)
B7 10.1201/9781003452652 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C4, in: Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1#C4. https://entrylib.org/p/2026.000006/C4
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