CLAIMSET Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
summary Pure multiplicative exponential smoothing (ETS) models are widely used for strictly positive time series, yet standard implementations assume normally distributed errors that can produce negative forecasts and unrealistic prediction intervals on low-volume data. This work shows that conventional point forecasts from pure multiplicative ETS models generally underestimate conditional expectations, with the bias increasing as the damping parameter, trend smoothing parameter, error variance, or forecast horizon grows. To preserve positivity, we formulate pure multiplicative ETS models with Log-Normal, Gamma, and Inverse Gaussian error distributions, deriving their concentrated log-likelihoods, scale estimators, and conditional moments. Simulation experiments and an electric vehicle charging case study demonstrate that information criteria reliably select among these distributions, and that positive distributions prevent negative prediction intervals without requiring transformation to logarithmic ETS models.
licence CC-BY-4.0; contributions C1, C3, C4, C5, C9, C10; highlights C1, C4, C5, C9, C10
declarations funding: none | competing interests: None declared.

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]
D8 [variable] percentage difference (PD): Relative deviation of the conventional point forecast $\hat{y}_{t+h}$ from the conditional expectation $\mu_{y,t+h}$, defined as $$PD = \frac{\mu_{y,t+h} - \hat{y}_{t+h}}{\mu_{y,t+h}}$$ lying in $[0, 1)$. [unit 1]
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]
D13 [method] ETS(M,Md,N): Pure multiplicative exponential smoothing model with damped multiplicative trend and no seasonality. [a kind of D1] [source B3]
D14 [data] Simulation experiment 1 synthetic series: 10,000 iterations of 100-observation time series simulated from $ETS(M,Md,N)$ with Inverse Gaussian errors across parameter grids $\sigma \in \{0.01, 0.05, 0.1, 0.5, 1\}$, $\alpha, \beta \in \{0, 0.1, 0.2, 0.5\}$, $\phi \in \{0.5, 0.9, 0.95, 1\}$, and horizons $h \in \{10, 50, 100\}$.
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]
D19 [design] Box-Cox odds regression procedure: Transformation of percentage difference $PD$ into odds $PD/(1-PD)$ followed by Box-Cox transformation ($\lambda = 0.146$) and linear regression on scaled parameters $\sigma$, horizon, $\alpha$, $\beta$, and $\phi$ using `greybox::alm` (`distribution='dbcnorm'`).
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
C1 [formal; principal; extension; support: formal_proof E1, cited B3, at pp. 21-23; assumes C12, C13; limitations L1; verification unchecked; verifiability open; planning exploratory]
  In general, the conventional multi-step point forecast $\hat{y}_{t+h}$ of a pure multiplicative [[D1:ETS(M,*,*)]] model does not coincide with the conditional arithmetic expectation $\mu_{y,t+h}$ or the conditional geometric expectation $\tilde{y}_{t+h}$, satisfying $\hat{y}_{t+h} \le \tilde{y}_{t+h} \le \mu_{y,t+h}$ when smoothing parameters are non-zero.
  scope: conditions: pure multiplicative ETS models with non-zero smoothing parameters and forecast horizon $h > 1$
  extends isbn 978-3-540-71918-2, at Chapter 2, pp. 21-23: Derives the explicit ordering between point forecasts, geometric conditional expectations, and arithmetic conditional expectations using logarithmic state space recursion.
C3 [descriptive; principal; support: simulation E2 (E2.T1) artefacts A3; assumes C12, C13; limitations L2; strength high (author); verifiability unverifiable; planning exploratory]
  For [[D13:ETS(M,Md,N)]] simulated on [[D14:Simulation experiment 1 synthetic series]] with an [[D6:Inverse Gaussian error distribution]], the [[D8:percentage difference (PD)]] between the conditional expectation and the point forecast increases with the damping parameter $\phi$, the trend smoothing parameter $\beta$, the forecast horizon $h$, the scale parameter $\sigma$, and the level smoothing parameter $\alpha$, with $\phi$ having the largest effect and $\alpha$ having the smallest effect among the scaled predictors.
  scope: conditions: $ETS(M,Md,N)$ simulated over $10,000$ iterations with Inverse Gaussian errors across parameter grids
  contradicts isbn 978-3-540-71918-2, at Chapter 2, p. 23: Shows that the difference between point forecasts and conditional expectations is non-negligible and grows substantially with horizon and parameter values.
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.
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.
C9 [descriptive; principal; support: simulation E5 (E5.T1, E5.T2) artefacts A3; assumes C12, C13; 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 $48$
C10 [descriptive; principal; support: simulation E5 (E5.T1, E5.T2) artefacts A3; assumes C12, C13; limitations L4; strength moderate (author); verifiability unverifiable; planning exploratory]
  Across all five data-generating processes in [[D15:Simulation experiment 2 synthetic series]], [[D10:Akram's log-ETS model]] achieves lower mean [[D11:scaled pinball score (SPS)]] over the 48-step horizon than pure multiplicative [[D3:ETS(M,Md,M)]] models with [[D7:Normal error distribution]], [[D6:Inverse Gaussian error distribution]], [[D4:Log-Normal error distribution]], [[D5:Gamma error distribution]], or [[D9:AICc]]-selected distributions, driven primarily by lower pinball loss in the central quantiles ($\tau \in [0.2, 0.8]$) while performing similarly in the distribution tails ($\tau < 0.2$ and $\tau > 0.8$).
  scope: conditions: $10,000$ simulated series of length $500$ per DGP, $99$ quantile levels from $0.01$ to $0.99$, horizon $48$
  extends doi 10.1111/j.1467-842X.2009.00555.x, at Section 4: Benchmarks Akram's logarithmic ETS model against pure multiplicative ETS with positive error distributions across full predictive 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 $E(1+\epsilon_t) = 1$.
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 $1 + \epsilon_t > 0$.
C2 [formal; supporting; extension; support: formal_proof E1; assumes C12; verification unchecked; verifiability open; planning exploratory]
  The conventional point forecast of a pure multiplicative [[D1:ETS(M,*,*)]] model coincides with both the conditional geometric and conditional arithmetic expectations only when $h=1$, when $h \le m$ for non-trended models with multiplicative seasonality, or for [[D2:ETS(M,N,N)]].
  scope: conditions: pure multiplicative ETS models with unit-mean i.i.d. multiplicative errors
C6 [formal; supporting; support: formal_proof E3; assumes C12; verification unchecked; 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 unchecked; 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 unchecked; 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]], while 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
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]] ($6,398.570$) than additive [[D12:ETS(A,Ad,A)]] with [[D7:Normal error distribution]] ($6,418.296$), and on a low-volume subset of the first 49 observations produces strictly positive 95% prediction intervals whereas the [[D7:Normal error distribution]] model produces negative lower prediction bounds.
  scope: conditions: $511$ daily EV charging observations (holdout $14$) and low-volume initial $49$ observations (holdout $14$)

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)
E2 [simulation; runs 10000] Monte Carlo simulation of $10,000$ iterations of $100$ observations from $ETS(M,Md,N)$ with Inverse Gaussian errors using `smooth::sim.es()`, followed by Box-Cox odds regression (`greybox::alm` with `distribution='dbcnorm'`) of percentage difference $PD$ on scaled parameters $\sigma$, horizon, $\alpha$, $\beta$, and $\phi$ (Section 3, Table 1).
  setup: studied D13, D6; cases D14; procedure D19; lens D8
  E2.T1 Box-Cox odds regression output for percentage difference $PD$ in Simulation Experiment 1 ($\lambda = 0.146$, $R^2 = 0.986$, $\text{df} = 534,884$).
  | row | variable | estimate (1) | std_error (1) | lower_0.05pct (1) | upper_99.95pct (1) |
  | r1 | (Intercept) | -9.222 | 0.006 | -9.242 | -9.202 |
  | r2 | sigma | 2.755 | 0.003 | 2.745 | 2.765 |
  | r3 | horizon | 3.656 | 0.003 | 3.646 | 3.667 |
  | r4 | alpha | 1.578 | 0.007 | 1.555 | 1.6 |
  | r5 | beta | 3.739 | 0.007 | 3.716 | 3.762 |
  | r6 | phi | 6.167 | 0.006 | 6.148 | 6.187 |
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)
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 S, e, c, t, i, o, n,  , 4, ., 2, ,,  , E, q, .,  , (, 4, ., 5, ); 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 S, e, c, t, i, o, n,  , 4, ., 2, ,,  , E, q, .,  , (, 4, ., 7, ); 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 D7, D6, D5, D4, D10; cases D15; procedure D20; lens 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
L1 [model_specification; affects C1; reduces_transferability; mitigation none] Conditional expectations for pure multiplicative ETS models other than $ETS(M,N,N)$ lack closed-form analytical expressions and require Monte Carlo simulation.
L2 [model_specification; affects C3; reduces_transferability; mitigation none] Simulation experiment 1 evaluates point forecast bias exclusively on $ETS(M,Md,N)$ with Inverse Gaussian errors and known true parameters.
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.

OPEN QUESTIONS
O1 [from C1, C3] How can a pure multiplicative state space model be formulated so that it retains positive support without exhibiting explosive multiplicative trends in place of $ETS(M,M,*)$?

ARTEFACTS
A0 [manuscript] The published article, IMA Journal of Management Mathematics (2023).
A1 [code] R simulation and estimation code using smooth package v3.2.0 (shared on reasonable request to the corresponding author).
A2 [data] Daily electric vehicle charging demand dataset (511 observations) analysed in Section 6 (restricted by non-disclosure agreement).
A3 [table] Full simulation results tables (Table 1 from Section 3 and Tables 4-5 from Section 5) exported as CSV.

REFERENCES
B1 10.1093/imaman/dpad028 (doi)
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)

SCORES
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.
Reliability of the paper: 52 (low), the mean of its 6 principal claims; the lowest is C9 at 31.
C1: reliability 76 (medium); novelty: extends earlier work, by the authors
C2: reliability 72 (medium); novelty: extends earlier work, by the authors
C3: reliability 36 (low); novelty: new, by the authors
C4: reliability 62 (medium); novelty: extends earlier work, by the authors · in question: close match on EntryLib not acknowledged
C5: reliability 72 (medium); novelty: extends earlier work, by the authors · in question: close match on EntryLib not acknowledged
C6: reliability 72 (medium); novelty: new, by the authors
C7: reliability 74 (medium); novelty: refutes earlier work, by the authors
C8: reliability 72 (medium); novelty: extends earlier work, by the authors
C9: reliability 31 (low); novelty: new, by the authors · in question: close match on EntryLib not acknowledged
C10: reliability 36 (low); novelty: new, by the authors · in question: close match on EntryLib not acknowledged
C11: reliability 46 (low); novelty: new, by the authors
Open problems in the principal claims and what they rest on:
P1 [checkable; up to -45; C3, C9, C10, C11] Verifiability: unverifiable. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
P2 [support; up to -25; C1, C2, C3, C4, C5, C6, C7, C8, C9, C10] 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.
P3 [open; up to -10; C4] A principal claim that declares no limitation. To remove it: Declare what could bias, reverse or limit the claim within its scope, each as a limitation that affects it.
P4 [plan; up to -5; C3, C4, C9, C10] A principal claim reported without a plan recorded beforehand. To remove it: Plan the measurement in the lab record or a registration before running it, or publish the lab record that shows every run.
P5 [context; up to -5; C9] A principal claim related to no earlier work. To remove it: Relate the claim to the earlier work it extends, addresses or departs from.
P6 [open; up to -3; C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11] 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.
P7 [open; up to -1; C1] Limitation L1 (reduces transferability, not mitigated): Conditional expectations for pure multiplicative ETS models other than $ETS(M,N,N)$ lack closed-form analytical expressions and require Monte Carlo simulation. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
P8 [open; up to -1; C3] Limitation L2 (reduces transferability, not mitigated): Simulation experiment 1 evaluates point forecast bias exclusively on $ETS(M,Md,N)$ with Inverse Gaussian errors and known true parameters. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
P9 [open; up to -1; C9, C10] Limitation L4 (reduces transferability, not mitigated): 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. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

HOW TO CITE
This claimset: 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
One claim of it: by its identifier, as entrylib:2026.000006@v1#C1 for claim C1. Each principal claim has its own page, as https://entrylib.org/p/2026.000006/C1.