CLAIM C9 of entrylib:2026.000006 Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
licence CC-BY-4.0; contributions C9

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]
D2 [method] ETS(M,N,N): Pure multiplicative exponential smoothing model with no trend and no seasonality, defined by $$\begin{aligned} y_t &= l_{t-1}(1 + \epsilon_t) \\ l_t &= l_{t-1}(1 + \alpha \epsilon_t) \end{aligned}$$ where $l_t$ is the level, $\alpha \in [0,1]$ is the smoothing parameter, and $1+\epsilon_t$ is the positive multiplicative error term. [a kind of D1] [source B3]
D3 [method] ETS(M,Md,M): Pure multiplicative exponential smoothing model with damped multiplicative trend and multiplicative seasonality, defined by $$\begin{aligned} y_t &= l_{t-1} b_{t-1}^{\phi} s_{t-m} (1 + \epsilon_t) \\ l_t &= l_{t-1} b_{t-1}^{\phi} (1 + \alpha \epsilon_t) \\ b_t &= b_{t-1}^{\phi} (1 + \beta \epsilon_t) \\ s_t &= s_{t-m} (1 + \gamma \epsilon_t) \end{aligned}$$ where $\phi$ is the dampening parameter, $\alpha, \beta, \gamma$ are smoothing parameters, and $m$ is the seasonal lag. [a kind of D1] [source B3]
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]
D9 [variable] AICc: Corrected Akaike Information Criterion used to select among ETS component structures and error distributions. [unit 1] [source B6]
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]
D11 [variable] scaled pinball score (SPS): Quantile forecast accuracy measure scaled by the in-sample mean $\bar{y}$, defined for probability level $\tau$ over horizon $h$ as $$SPS = \frac{1}{\bar{y}} \left( (1-\tau) \sum_{y_{t+j} < \hat{q}_{\tau,t+j}} (\hat{q}_{\tau,t+j} - y_{t+j}) + \tau \sum_{y_{t+j} \ge \hat{q}_{\tau,t+j}} (y_{t+j} - \hat{q}_{\tau,t+j}) \right).$$ [unit 1]
D12 [method] ETS(A,Ad,A): Pure additive exponential smoothing model with additive error, damped additive trend, and additive seasonality under Normal errors. [source B3]
D15 [data] Simulation experiment 2 synthetic series: 10,000 simulated time series of length 500 generated for each of five data-generating processes ($ETS(M,Md,M)$ with Normal, Inverse Gaussian, Gamma, Log-Normal errors, and Akram's exponentiated $ETS(A,Ad,A)$) with $\sigma=0.1$, $\alpha=0.1$, $\beta=0.05$, $\gamma=0.2$, $\phi=0.95$, $l_0=100$, $b_0=1$, and holdout horizon $48$.
D16 [data] Daily EV charging dataset: 511 daily aggregated observations of electricity used by electric vehicles in a geographical area starting from low volume with 7-day multiplicative seasonality, holding out the last 14 observations.
D17 [variable] quantile bias: Mean across quantile levels $\tau \in \{0.01, 0.02, \dots, 0.99\}$ of empirical quantile coverage minus nominal target level $\tau$. [unit 1]
D18 [variable] quantile precision: Mean across quantile levels $\tau \in \{0.01, 0.02, \dots, 0.99\}$ of the absolute difference between empirical quantile coverage and nominal target level $\tau$. [unit 1]
D20 [design] 48-step quantile forecast simulation procedure: Estimation of $ETS(M,Md,M)$ with each candidate distribution, AICc-based automatic selection (`smooth::auto.adam`), and Akram's log-ETS model on 500 observations, evaluating 99 quantile levels ($\tau = 0.01, \dots, 0.99$) over a 48-step holdout.
D21 [design] two-dimensional AICc selection procedure: Simultaneous selection of pure multiplicative ETS component structure and error distribution by minimising $AICc$ via `smooth::auto.adam` with a 14-day holdout evaluation.

CLAIMS
C9 [descriptive; principal; support: simulation E5 (E5.T1.r1, E5.T1.r2, E5.T1.r3, E5.T1.r4, E5.T1.r6, E5.T2.r1, E5.T2.r2, E5.T2.r3, E5.T2.r4, E5.T2.r6) artefacts A3; assumes C12, C13; uses C4, C5, C11; limitations L4; strength moderate (author); verifiability unverifiable; planning exploratory]
  Selecting among [[D7:Normal error distribution]], [[D6:Inverse Gaussian error distribution]], [[D4:Log-Normal error distribution]], and [[D5:Gamma error distribution]] in [[D3:ETS(M,Md,M)]] via [[D9:AICc]] achieves mean [[D11:scaled pinball score (SPS)]], [[D17:quantile bias]], and [[D18:quantile precision]] comparable to specifying the true data-generating distribution across [[D15:Simulation experiment 2 synthetic series]].
  scope: conditions: $10,000$ simulated series of length $500$ per DGP with standard deviation $\sigma = 0.1$ and forecast horizon $48$
  extends doi 10.1080/03610927808837599, at Section 1: Applies AICc model selection across competing positive error distributions within pure multiplicative ETS models.
  supports claimset entrylib:2026.000003@v4#C17, own earlier work: Evaluates fully observed pure multiplicative ETS models across Gamma, Log-Normal, and Inverse Gaussian distributions, whereas entrylib:2026.000003#C17 derives moment equations under unobserved periods.
C4 [methodological; principal; extension; support: formal_proof E3, cited B4, at Section 2; assumes C12, C13; uses C8; limitations L6; strength high (author); verification agent_checked; 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.
  supports claimset entrylib:2026.000003@v4#C17, own earlier work: Provides the conventional multiplicative ETS formulation with Gamma, Log-Normal, and Inverse Gaussian errors subsequently extended to unobserved periods in entrylib:2026.000003.
C5 [formal; principal; extension; support: formal_proof E3; assumes C12; uses C6, C7; verification agent_checked; 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.
  supports claimset entrylib:2026.000003@v4#C17, own earlier work: Establishes the full-sample concentrated log-likelihoods for multiplicative ETS distributions later generalised to intermittent/missing-data settings in entrylib:2026.000003.
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 $1 + \epsilon_t > 0$.
C6 [formal; supporting; support: formal_proof E3; assumes C12; verification agent_checked; verifiability open; planning exploratory]
  The maximum likelihood estimator of the scale parameter $\sigma^2$ in [[D1:ETS(M,*,*)]] has closed form $\hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^T \epsilon_t^2$ for the [[D7:Normal error distribution]] and $\hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^T \frac{\epsilon_t^2}{1+\epsilon_t}$ for the [[D6:Inverse Gaussian error distribution]], whereas the [[D5:Gamma error distribution]] has no closed-form MLE under $\xi = \sigma^{-2}$ and uses $\hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^T \epsilon_t^2$ via the method of moments.
  scope: conditions: sample size $T$ with unit-mean multiplicative error distributions
C7 [formal; supporting; refutation; support: formal_proof E3, cited B2, own earlier work; assumes C12; limitations L3; verification agent_checked; verifiability open; planning exploratory]
  Under the unit-mean restriction $\mu = -\sigma^2/2$, the concentrated log-likelihood of [[D1:ETS(M,*,*)]] with a [[D4:Log-Normal error distribution]] is non-convex and its maximum likelihood scale estimator is given by the corrected formula in `entrylib:2026.000003@v4#C7` rather than the quadratic root formula printed in Section 4.1 of the 2023 published article.
  scope: conditions: $ETS(M,*,*)$ models with Log-Normal error distribution constrained to $E(1 + \epsilon_t) = 1$
  contradicts doi 10.1093/imaman/dpad028, at Section 4.1, pp. 9-10, own earlier work: Author erratum: the quadratic root expression for the Log-Normal scale parameter in Section 4.1 of the 2023 article was superseded by the corrected derivation in entrylib:2026.000003@v4#C7.
  extends claimset entrylib:2026.000003@v4#C7, own earlier work: References the corrected Log-Normal scale MLE formula.
C8 [formal; supporting; extension; support: formal_proof E4; assumes C12, C13; verification agent_checked; verifiability open; planning exploratory]
  Among pure multiplicative [[D1:ETS(M,*,*)]] models, [[D2:ETS(M,N,N)]] is the only model with closed-form conditional $h$-steps-ahead mean $E(l_{t+h}|t)=l_t$ and variance $V(l_{t+h}|t)=l_t^2((1+\alpha^2\sigma^2)^h-1)$ across [[D7:Normal error distribution]], [[D6:Inverse Gaussian error distribution]], [[D4:Log-Normal error distribution]], and [[D5:Gamma error distribution]], requiring $\alpha \in [0,1]$ to ensure positivity of the level.
  scope: conditions: $ETS(M,N,N)$ with $\alpha \in [0, 1]$ and unit-mean error distributions
  extends isbn 978-3-540-71918-2, at Chapter 6: Extends the closed-form ETS(M,N,N) conditional mean and variance formulas from Normal errors to Inverse Gaussian, Log-Normal, and Gamma distributions.
C11 [descriptive; supporting; support: empirical_quant E6 (E6.R1, E6.R2); assumes C12, C13; limitations L5; strength low (author); verifiability unverifiable; planning exploratory]
  On the [[D16:Daily EV charging dataset]], [[D3:ETS(M,Md,M)]] with a [[D5:Gamma error distribution]] achieves a lower [[D9:AICc]] ($6398.57$) than additive [[D12:ETS(A,Ad,A)]] with [[D7:Normal error distribution]] ($6418.296$), and on the initial $49$-observation low-volume subsample generates strictly positive $95\%$ prediction intervals unlike the [[D7:Normal error distribution]] model.
  scope: conditions: 511 daily EV charging observations (holdout 14) and low-volume initial 49 observations (holdout 14) at 95% prediction interval level

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 D1, C12; 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 E1.1, C12; 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 E1.2, C13; 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 D4, D5, D6, D7, C12; moment_constraint)
  E3.2 Substitute parameter restrictions into probability density functions and Jacobian terms to obtain concentrated log-likelihood functions. (by E3.1, D1; 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 E3.2, B2; maximum_likelihood_estimation)
E4 [formal; foundation classical_probability] Analytical derivation of closed-form conditional $h$-steps-ahead mean and variance, one-step-ahead predictive distributions via scalability, and smoothing parameter domain restrictions for $ETS(M,N,N)$ (Section 4.2, Eqs. 4.2-4.7).
  assumes C12, C13
  E4.1 Expand level recursion $l_{t+h} = l_t \prod_{j=1}^h (1 + \alpha \epsilon_{t+j})$ under i.i.d. unit-mean errors to derive $E(l_{t+h}|t) = l_t$ and $V(l_{t+h}|t) = l_t^2((1 + \alpha^2 \sigma^2)^h - 1)$. (by D2, C12; independent_product_moments)
  E4.2 Enforce positivity of level update $1 + \alpha \epsilon_t > 0$ for all $1 + \epsilon_t > 0$ to establish $\alpha \in [0, 1]$. (by E4.1, C13; positivity_bound)
E5 [simulation; runs 10000] Monte Carlo simulation of $10,000$ series of length $500$ for each of five DGPs ($ETS(M,Md,M)$ with Normal, Inverse Gaussian, Gamma, Log-Normal, and Akram's exponentiated $ETS(A,Ad,A)$), evaluating 48-step quantile forecasts across 99 probability levels via scaled pinball score, quantile bias, and quantile precision (Section 5, Tables 4-5).
  setup: studied D3, D7, D6, D5, D4, D10; cases D15; procedure D20; lens D9, D11, D17, D18
  E5.T1 Mean scaled pinball scores ($SPS$) across $10,000$ simulated series for applied models (rows) against data-generating processes (columns) over forecast horizon $48$.
  | row | applied_model | D11 {DGP: Normal} (1) | D11 {DGP: Inverse Gaussian} (1) | D11 {DGP: Gamma} (1) | D11 {DGP: Log-Normal} (1) | D11 {DGP: Akram} (1) |
  | r1 | Normal | 5.255 | 5.204 | 5.193 | 9.417 | 5.192 |
  | r2 | Inverse Gaussian | 5.3 | 5.221 | 5.224 | 9.464 | 5.229 |
  | r3 | Gamma | 5.287 | 5.192 | 5.231 | 9.388 | 5.214 |
  | r4 | Log-Normal | 5.215 | 5.295 | 5.224 | 9.423 | 5.22 |
  | r5 | Akram | 5.11 | 5.052 | 5.074 | 9.18 | 5.089 |
  | r6 | Auto (AICc) | 5.258 | 5.175 | 5.208 | 9.332 | 5.178 |
  E5.T2 Quantile bias and quantile precision across 99 quantile levels ($0.01$ to $0.99$) for applied models against data-generating processes.
  | row | applied_model | D17 {DGP: Normal} (1) | D17 {DGP: Inverse Gaussian} (1) | D17 {DGP: Gamma} (1) | D17 {DGP: Log-Normal} (1) | D17 {DGP: Akram} (1) | D18 {DGP: Normal} (1) | D18 {DGP: Inverse Gaussian} (1) | D18 {DGP: Gamma} (1) | D18 {DGP: Log-Normal} (1) | D18 {DGP: Akram} (1) |
  | r1 | Normal | -0.002 | 0.003 | 0.011 | 0.007 | -0.01 | 0.007 | 0.011 | 0.005 | 0.009 | 0.01 |
  | r2 | Inverse Gaussian | -0.004 | 0.009 | 0.001 | 0.001 | -0.013 | 0.009 | 0.009 | 0.006 | 0.009 | 0.015 |
  | r3 | Gamma | -0.004 | 0.008 | 0.001 | 0.003 | -0.012 | 0.011 | 0.009 | 0.005 | 0.008 | 0.014 |
  | r4 | Log-Normal | -0.005 | 0.008 | 0 | 0.001 | -0.013 | 0.008 | 0.004 | 0.008 | 0.004 | 0.014 |
  | r5 | Akram | 0 | 0.004 | 0.01 | 0.006 | -0.007 | 0.004 | 0.01 | 0.005 | 0.008 | 0.007 |
  | r6 | Auto (AICc) | -0.003 | 0.008 | 0 | 0.002 | -0.012 | 0.008 | 0.004 | 0.009 | 0.007 | 0.013 |
E6 [quant; design benchmark; data_origin secondary; n 511] Empirical application of automatic pure multiplicative ETS selection via `smooth::auto.adam()` to $511$ daily EV charging observations (holdout $14$) and a low-volume initial $49$-observation subsample (holdout $14$) comparing Gamma and Normal error distributions (Section 6, Figs. 4-6).
  setup: studied D3, D12, D5, D7; cases D16; procedure D21; lens D9
  E6.R1 D9 = 6398.57 1 n 497 {model: D3, distribution: D5}
  E6.R2 D9 = 6418.296 1 n 497 {model: D12, distribution: D7}

LIMITATIONS
L3 [model_specification; affects C7; reduces_precision; mitigation none] The Log-Normal concentrated log-likelihood under unit-mean restriction is non-convex, creating potential local optima during numerical optimisation.
L4 [model_specification; affects C9, C10; reduces_transferability; mitigation none] Simulation experiment 2 uses a single fixed parameter vector with error standard deviation $\sigma = 0.1$, where multiplicative errors remain small enough that Normal and logarithmic approximations behave regularly.
L5 [sampling; affects C11; reduces_transferability; mitigation none] The real-life demonstration relies on a single proprietary daily electric vehicle charging series that cannot be publicly shared under a non-disclosure agreement.
L6 [model_specification; affects C4; reduces_transferability; mitigation none] Multi-step conditional predictive distributions for $ETS(M,*,*)$ models with Log-Normal, Gamma, or Inverse Gaussian errors have no closed form because none of these distributions is closed under both convolution and product, requiring simulation.

ARTEFACTS
A3 [table] Full simulation results tables (Table 1 from Section 3 and Tables 4-5 from Section 5) exported as CSV.

REFERENCES
B2 entrylib:2026.000003@v4#C7 (claimset)
B3 978-3-540-71918-2 (isbn)
B4 10.1111/j.1467-842X.2009.00555.x (doi)
B6 10.1080/03610927808837599 (doi)
B7 10.1201/9781003452652 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C9, in: Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v2#C9. https://entrylib.org/p/2026.000006/C9
The claimset it is from: Svetunkov, I. (guarantor) (2026). Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v2. https://entrylib.org/p/2026.000006@v2