Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models: claim C5

The claim

Support 72 C5 · formal · extends earlier work · novelty: extends earlier work, by the authors · in question: close match on EntryLib not acknowledged

Concentrated log-likelihood functions for ETS(M,*,*) models under the unit-mean restriction $E(1+\epsilon_t)=1$ take closed forms for Normal error distribution $\mathcal{N}(1,\sigma^2)$ , Inverse Gaussian error distribution $\mathcal{IG}(1,\sigma^2)$ , Log-Normal error distribution $\log\mathcal{N}(-\sigma^2/2,\sigma^2)$ , and Gamma error distribution $\Gamma(\sigma^{-2},\sigma^2)$ error distributions.

Scope: conditions: $ETS(M,*,*)$ models estimated via maximum likelihood on strictly positive observations · Proof: E3 · Assumes: C12 · Extends: 978-3-540-71918-2 · Verification: not checked

Evidence

E3 · formal · cited by C5

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 · Assumes: C12

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. (moment_constraint; from S, e, c, t, i, o, n, , 4, ., 1, ,, , T, a, b, l, e, , 2)
Substitute parameter restrictions into probability density functions and Jacobian terms to obtain concentrated log-likelihood functions. (likelihood_derivation; from S, e, c, t, i, o, n, , 4, ., 1, ,, , T, a, b, l, e, , 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. (maximum_likelihood_estimation; from 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)

Assumptions

C12 · statistical · unchecked

The multiplicative error term $(1+\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\epsilon_t) = 1$ . Rested on by C5, E3.

Definitions

Methods 1

D1 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: Svetunkov (2023).

Concepts 4

D4 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 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 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 Normal error distribution: Classical Gaussian distribution on $(-\infty, \infty)$ applied to $1+\epsilon_t$ with mean $\mu = 1$ and variance $\sigma^2$ . Source: 978-3-540-71918-2.

Context

Extends 978-3-540-71918-2 · from C5: Generalises the Normal multiplicative ETS log-likelihood to Inverse Gaussian, Log-Normal, and Gamma distributions.

References

978-3-540-71918-2 978-3-540-71918-2 Used by: D7
Svetunkov (2023) Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM). Chapman and Hall/CRC. doi:10.1201/9781003452652 Used by: D1

Score

Scores are computed by rules from the claimset file (EntryLib rubric 0.4): they check the record, not the science, and may change as the rubric and reviews develop. The rubric.

C5: 72, Partly supported

Novelty: extends earlier work, by the authors · in question: close match on EntryLib not acknowledged

−25 Evidence Rests on an unchecked proof. To remove it: Support the claim with evidence the claimset records, citing the result entries or table rows it reports, or have its proof checked.
−3 Open problems Assumption C12 is not checked: The multiplicative error term $(1+\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\epsilon_t) = 1$ . To remove it: Test the assumption, or analyse how sensitive the claim is to it.

Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models · C5