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Version 1 · 12 November 2026 · Management Science and Operations Research

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

Responsible researcher: Ivan Svetunkov (0000-0001-7826-0281)

Written with: AI agentGemini · claimset preparation

CC-BY-4.0 · Keywords: exponential smoothing, state space models, multiplicative error models, positive time series forecasting, predictive distributions

Converted by its authors from their paper: The published article, IMA Journal of Management Mathematics (2023)..

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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.

Highlights

Context

Extends 978-3-540-71918-2 · from C1
Derives the explicit ordering between point forecasts, geometric conditional expectations, and arithmetic conditional expectations using logarithmic state space recursion.
Contradicts 978-3-540-71918-2 · from C3
Shows that the difference between point forecasts and conditional expectations is non-negligible and grows substantially with horizon and parameter values.
Extends doi:10.1111/j.1467-842X.2009.00555.x · from C4
Implements Log-Normal, Gamma, and Inverse Gaussian distributions directly within original ETS(M,*,*)ETS(M,*,*) rather than sidestepping the taxonomy via logarithmic ETS.
Extends 978-3-540-71918-2 · from C5
Generalises the Normal multiplicative ETS log-likelihood to Inverse Gaussian, Log-Normal, and Gamma distributions.
Contradicts doi:10.1093/imaman/dpad028 · own earlier work · from C7
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 entrylib:2026.000003@v4#C7 · own earlier work · from C7
References the corrected Log-Normal scale MLE formula.
Extends doi:10.1111/j.1467-842X.2009.00555.x · from C10
Benchmarks Akram's logarithmic ETS model against pure multiplicative ETS with positive error distributions across full predictive distributions.

Definitions

Methods 6

D1 ETS(M,*,*): Pure multiplicative exponential smoothing state space model family written in logarithmic state space form as log⁡yt=w′log⁡(vt−l)+log⁡(1+ϵt)log⁡vt=Flog⁡vt−l+log⁡(1k+gϵt)\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 y^t+h=exp⁡((w1′F1h−1+wm′Fm⌈h/m⌉−1)log⁡vt)\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).

D2 · a kind of D1 ETS(M,N,N): Pure multiplicative exponential smoothing model with no trend and no seasonality, defined by yt=lt−1(1+ϵt)lt=lt−1(1+αϵt)\begin{aligned} y_t &= l_{t-1}(1 + \epsilon_t) \\ l_t &= l_{t-1}(1 + \alpha \epsilon_t) \end{aligned} where ltl_t is the level, α∈[0,1]\alpha \in [0,1] is the smoothing parameter, and 1+ϵt1+\epsilon_t is the positive multiplicative error term. Source: 978-3-540-71918-2.

D3 · a kind of D1 ETS(M,Md,M): Pure multiplicative exponential smoothing model with damped multiplicative trend and multiplicative seasonality, defined by yt=lt−1bt−1ϕst−m(1+ϵt)lt=lt−1bt−1ϕ(1+αϵt)bt=bt−1ϕ(1+βϵt)st=st−m(1+γϵt)\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 ϕ\phi is the dampening parameter, α,β,γ\alpha, \beta, \gamma are smoothing parameters, and mm is the seasonal lag. Source: 978-3-540-71918-2.

D13 · a kind of D1 ETS(M,Md,N): Pure multiplicative exponential smoothing model with damped multiplicative trend and no seasonality. Source: 978-3-540-71918-2.

D10 Akram's log-ETS model: Additive error ETS model ETS(A,Ad,A)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: Akram et al. (2009).

D12 ETS(A,Ad,A): Pure additive exponential smoothing model with additive error, damped additive trend, and additive seasonality under Normal errors. Source: 978-3-540-71918-2.

Data 3

D14 Simulation experiment 1 synthetic series: 10,000 iterations of 100-observation time series simulated from ETS(M,Md,N)ETS(M,Md,N) with Inverse Gaussian errors across parameter grids σ∈{0.01,0.05,0.1,0.5,1}\sigma \in \{0.01, 0.05, 0.1, 0.5, 1\}, α,β∈{0,0.1,0.2,0.5}\alpha, \beta \in \{0, 0.1, 0.2, 0.5\}, ϕ∈{0.5,0.9,0.95,1}\phi \in \{0.5, 0.9, 0.95, 1\}, and horizons h∈{10,50,100}h \in \{10, 50, 100\}.

D15 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)ETS(M,Md,M) with Normal, Inverse Gaussian, Gamma, Log-Normal errors, and Akram's exponentiated ETS(A,Ad,A)ETS(A,Ad,A)) with σ=0.1\sigma=0.1, α=0.1\alpha=0.1, β=0.05\beta=0.05, γ=0.2\gamma=0.2, ϕ=0.95\phi=0.95, l0=100l_0=100, b0=1b_0=1, and holdout horizon 4848.

D16 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.

Variables and measures 5

D8 percentage difference (PD): Relative deviation of the conventional point forecast y^t+h\hat{y}_{t+h} from the conditional expectation μy,t+h\mu_{y,t+h}, defined as PD=μy,t+h−y^t+hμy,t+hPD = \frac{\mu_{y,t+h} - \hat{y}_{t+h}}{\mu_{y,t+h}} lying in [0,1)[0, 1).

D9 AICc: Corrected Akaike Information Criterion used to select among ETS component structures and error distributions. Source: doi:10.1080/03610927808837599.

D11 scaled pinball score (SPS): Quantile forecast accuracy measure scaled by the in-sample mean y¯\bar{y}, defined for probability level τ\tau over horizon hh as SPS=1y¯((1−τ)∑yt+j<q^τ,t+j(q^τ,t+j−yt+j)+τ∑yt+j≥q^τ,t+j(yt+j−q^τ,t+j)).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).

D17 quantile bias: Mean across quantile levels τ∈{0.01,0.02,…,0.99}\tau \in \{0.01, 0.02, \dots, 0.99\} of empirical quantile coverage minus nominal target level τ\tau.

D18 quantile precision: Mean across quantile levels τ∈{0.01,0.02,…,0.99}\tau \in \{0.01, 0.02, \dots, 0.99\} of the absolute difference between empirical quantile coverage and nominal target level τ\tau.

Study design 3

D19 Box-Cox odds regression procedure: Transformation of percentage difference PDPD into odds PD/(1−PD)PD/(1-PD) followed by Box-Cox transformation (λ=0.146\lambda = 0.146) and linear regression on scaled parameters σ\sigma, horizon, α\alpha, β\beta, and ϕ\phi using greybox::alm (distribution='dbcnorm').

D20 48-step quantile forecast simulation procedure: Estimation of ETS(M,Md,M)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 (τ=0.01,…,0.99\tau = 0.01, \dots, 0.99) over a 48-step holdout.

D21 two-dimensional AICc selection procedure: Simultaneous selection of pure multiplicative ETS component structure and error distribution by minimising AICcAICc via smooth::auto.adam with a 14-day holdout evaluation.

Concepts 4

D4 Log-Normal error distribution: Probability distribution on (0,∞)(0, \infty) applied to 1+ϵt1+\epsilon_t with location μ=−σ2/2\mu = -\sigma^2/2 and scale σ2\sigma^2 so that E(1+ϵt)=1E(1+\epsilon_t) = 1 and V(1+ϵt)=eσ2−1V(1+\epsilon_t) = e^{\sigma^2} - 1.

D5 Gamma error distribution: Probability distribution on (0,∞)(0, \infty) applied to 1+ϵt1+\epsilon_t with shape ξ=σ−2\xi = \sigma^{-2} and scale σ2\sigma^2 so that E(1+ϵt)=1E(1+\epsilon_t) = 1 and V(1+ϵt)=σ2V(1+\epsilon_t) = \sigma^2.

D6 Inverse Gaussian error distribution: Probability distribution on (0,∞)(0, \infty) applied to 1+ϵt1+\epsilon_t with location μ=1\mu = 1 and scale σ2\sigma^2 so that E(1+ϵt)=1E(1+\epsilon_t) = 1 and V(1+ϵt)=σ2V(1+\epsilon_t) = \sigma^2.

D7 Normal error distribution: Classical Gaussian distribution on (−∞,∞)(-\infty, \infty) applied to 1+ϵt1+\epsilon_t with mean μ=1\mu = 1 and variance σ2\sigma^2. Source: 978-3-540-71918-2.

Assumptions

C12 · statistical · unchecked

The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. Rested on by C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, E1, E3, E4.

C13 · methodological · tested

Smoothing parameters α,β,γ\alpha, \beta, \gamma in pure multiplicative ETS models lie within [0,1][0, 1] so that state update terms (1+gϵt)(1 + g\epsilon_t) remain strictly positive whenever 1+ϵt>01 + \epsilon_t > 0. Rested on by C1, C3, C4, C8, C9, C10, C11, E1, E4.

Principal claims

Support 76C1 · formal · extends earlier work · novelty: extends earlier work, by the authors

In general, the conventional multi-step point forecast y^t+h\hat{y}_{t+h} of a pure multiplicative ETS(M,*,*) model does not coincide with the conditional arithmetic expectation μy,t+h\mu_{y,t+h} or the conditional geometric expectation y~t+h\tilde{y}_{t+h}, satisfying y^t+h≤y~t+h≤μy,t+h\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>1h > 1 · Proof: E1 · Cited: 978-3-540-71918-2, pp. 21-23 · Assumes: C12, C13 · Limitations: L1 · Extends: 978-3-540-71918-2 · Verification: not checked

Support 36C3 · descriptive · novelty: new, by the authors

For ETS(M,Md,N) simulated on Simulation experiment 1 synthetic series with an Inverse Gaussian error distribution, the percentage difference (PD) between the conditional expectation and the point forecast increases with the damping parameter ϕ\phi, the trend smoothing parameter β\beta, the forecast horizon hh, the scale parameter σ\sigma, and the level smoothing parameter α\alpha, with ϕ\phi having the largest effect and α\alpha having the smallest effect among the scaled predictors.

Scope: conditions: ETS(M,Md,N)ETS(M,Md,N) simulated over 10,00010,000 iterations with Inverse Gaussian errors across parameter grids · Simulation: E2 (T1), artefacts A3 · Assumes: C12, C13 · Limitations: L2 · Contradicts: 978-3-540-71918-2

Support 62C4 · methodological · extends earlier work · novelty: extends earlier work, by the authors · in question: close match on EntryLib not acknowledged

Log-Normal error distribution, Gamma error distribution, and Inverse Gaussian error distribution distributions defined on (0,∞)(0, \infty) can be applied directly to the multiplicative error term (1+ϵt)(1+\epsilon_t) of conventional ETS(M,*,*) models without transforming the model into Akram's log-ETS model, provided E(1+ϵt)=1E(1+\epsilon_t)=1.

Scope: conditions: strictly positive time series modelled within the original ETS(M,*,*)ETS(M,*,*) taxonomy · Proof: E3 · Cited: Akram et al. (2009), Section 2 · Assumes: C12, C13 · Extends: doi:10.1111/j.1467-842X.2009.00555.x · Verification: not checked

Support 72C5 · 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+ϵt)=1E(1+\epsilon_t)=1 take closed forms for Normal error distribution 𝒩(1,σ2)\mathcal{N}(1,\sigma^2), Inverse Gaussian error distribution ℐ𝒢(1,σ2)\mathcal{IG}(1,\sigma^2), Log-Normal error distribution log⁡𝒩(−σ2/2,σ2)\log\mathcal{N}(-\sigma^2/2,\sigma^2), and Gamma error distribution Γ(σ−2,σ2)\Gamma(\sigma^{-2},\sigma^2) error distributions.

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

Support 31C9 · descriptive · novelty: new, by the authors · in question: close match on EntryLib not acknowledged

Selecting among Normal error distribution, Inverse Gaussian error distribution, Log-Normal error distribution, and Gamma error distribution in ETS(M,Md,M) via AICc achieves mean scaled pinball score (SPS), quantile bias, and quantile precision comparable to specifying the true data-generating distribution across Simulation experiment 2 synthetic series.

Scope: conditions: 10,00010,000 simulated series of length 500500 per DGP with standard deviation σ=0.1\sigma = 0.1 and forecast horizon 4848 · Simulation: E5 (T1, T2), artefacts A3 · Assumes: C12, C13 · Limitations: L4

Support 36C10 · descriptive · novelty: new, by the authors · in question: close match on EntryLib not acknowledged

Across all five data-generating processes in Simulation experiment 2 synthetic series, Akram's log-ETS model achieves lower mean scaled pinball score (SPS) over the 48-step horizon than pure multiplicative ETS(M,Md,M) models with Normal error distribution, Inverse Gaussian error distribution, Log-Normal error distribution, Gamma error distribution, or AICc-selected distributions, driven primarily by lower pinball loss in the central quantiles (τ∈[0.2,0.8]\tau \in [0.2, 0.8]) while performing similarly in the distribution tails (τ<0.2\tau < 0.2 and τ>0.8\tau > 0.8).

Scope: conditions: 10,00010,000 simulated series of length 500500 per DGP, 9999 quantile levels from 0.010.01 to 0.990.99, horizon 4848 · Simulation: E5 (T1, T2), artefacts A3 · Assumes: C12, C13 · Limitations: L4 · Extends: doi:10.1111/j.1467-842X.2009.00555.x

Supporting claims

Support 72C2 · formal · extends earlier work · novelty: extends earlier work, by the authors

The conventional point forecast of a pure multiplicative ETS(M,*,*) model coincides with both the conditional geometric and conditional arithmetic expectations only when h=1h=1, when h≤mh \le m for non-trended models with multiplicative seasonality, or for ETS(M,N,N).

Scope: conditions: pure multiplicative ETS models with unit-mean i.i.d. multiplicative errors · Proof: E1 · Assumes: C12 · Verification: not checked

Support 72C6 · formal · novelty: new, by the authors

The maximum likelihood estimator of the scale parameter σ2\sigma^2 in ETS(M,*,*) has closed form σ^2=1T∑t=1Tϵt2\hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^T \epsilon_t^2 for the Normal error distribution and σ^2=1T∑t=1Tϵt21+ϵt\hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^T \frac{\epsilon_t^2}{1+\epsilon_t} for the Inverse Gaussian error distribution, whereas the Gamma error distribution has no closed-form MLE under ξ=σ−2\xi = \sigma^{-2} and uses σ^2=1T∑t=1Tϵt2\hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^T \epsilon_t^2 via the method of moments.

Scope: conditions: sample size TT with unit-mean multiplicative error distributions · Proof: E3 · Assumes: C12 · Verification: not checked

Support 74C7 · formal · refutes earlier work · novelty: refutes earlier work, by the authors

Under the unit-mean restriction μ=−σ2/2\mu = -\sigma^2/2, the concentrated log-likelihood of ETS(M,*,*) with a 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,*,*)ETS(M,*,*) models with Log-Normal error distribution constrained to E(1+ϵt)=1E(1 + \epsilon_t) = 1 · Proof: E3 · Cited: entrylib:2026.000003@v4#C7, own earlier work · Assumes: C12 · Limitations: L3 · Contradicts: doi:10.1093/imaman/dpad028 · Extends: entrylib:2026.000003@v4#C7 · Verification: not checked

Support 72C8 · formal · extends earlier work · novelty: extends earlier work, by the authors

Among pure multiplicative ETS(M,*,*) models, ETS(M,N,N) is the only model with closed-form conditional hh-steps-ahead mean E(lt+h|t)=ltE(l_{t+h}|t)=l_t and variance V(lt+h|t)=lt2((1+α2σ2)h−1)V(l_{t+h}|t)=l_t^2((1+\alpha^2\sigma^2)^h-1) across Normal error distribution, Inverse Gaussian error distribution, Log-Normal error distribution, and Gamma error distribution, while requiring α∈[0,1]\alpha \in [0,1] to ensure positivity of the level.

Scope: conditions: ETS(M,N,N)ETS(M,N,N) with α∈[0,1]\alpha \in [0, 1] and unit-mean error distributions · Proof: E4 · Assumes: C12, C13 · Verification: not checked

Support 46C11 · descriptive · novelty: new, by the authors

On the Daily EV charging dataset, ETS(M,Md,M) with a Gamma error distribution achieves a lower AICc (6,398.5706,398.570) than additive ETS(A,Ad,A) with Normal error distribution (6,418.2966,418.296), and on a low-volume subset of the first 49 observations produces strictly positive 95% prediction intervals whereas the Normal error distribution model produces negative lower prediction bounds.

Scope: conditions: 511511 daily EV charging observations (holdout 1414) and low-volume initial 4949 observations (holdout 1414) · Data: E6 (R1, R2) · Assumes: C12, C13 · Limitations: L5

Evidence

E1 · formal · cited by C1, C2

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).

Foundation: classical_probability · Assumes: C12, C13

  1. Reformulate pure multiplicative ETS in logarithmic state space form: log⁡yt=w′log⁡(vt−l)+log⁡(1+ϵt)\log y_t = w' \log(v_{t-l}) + \log(1 + \epsilon_t) and log⁡vt=Flog⁡vt−l+log⁡(1k+gϵt)\log v_t = F \log v_{t-l} + \log(1_k + g \epsilon_t). (logarithmic_transformation; from S, e, c, t, i, o, n, , 2, ,, , E, q, ., , (, 2, ., 1, ))
  2. Expand hh-steps-ahead recursion into non-seasonal and seasonal components and take conditional expectation E(log⁡yt+h|t)E(\log y_{t+h}|t). (recursive_substitution; from S, e, c, t, i, o, n, , 2, ,, , E, q, s, ., , (, 2, ., 5, ), -, (, 2, ., 6, ))
  3. Apply Jensen's inequality and non-negativity of gg to show point forecast y^t+h≤y~t+h≤μy,t+h\hat{y}_{t+h} \le \tilde{y}_{t+h} \le \mu_{y,t+h}, with equality only at h=1h=1, h≤mh \le m without trend, or for ETS(M,N,N)ETS(M,N,N). (jensens_inequality; from S, e, c, t, i, o, n, , 2, ,, , E, q, s, ., , (, 2, ., 8, ), -, (, 2, ., 1, 1, ))

E2 · simulation · cited by C3

Monte Carlo simulation of 10,00010,000 iterations of 100100 observations from ETS(M,Md,N)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 PDPD on scaled parameters σ\sigma, horizon, α\alpha, β\beta, and ϕ\phi (Section 3, Table 1).

Setup: studied ETS(M,Md,N), Inverse Gaussian error distribution · cases Simulation experiment 1 synthetic series · procedure Box-Cox odds regression procedure · lens percentage difference (PD)

E2.T1 · Box-Cox odds regression output for percentage difference PDPD in Simulation Experiment 1 (λ=0.146\lambda = 0.146, R2=0.986R^2 = 0.986, df=534,884\text{df} = 534,884).
Rowvariableestimate (1)std_error (1)lower_0.05pct (1)upper_99.95pct (1)
r1(Intercept)-9.2220.006-9.242-9.202
r2sigma2.7550.0032.7452.765
r3horizon3.6560.0033.6463.667
r4alpha1.5780.0071.5551.6
r5beta3.7390.0073.7163.762
r6phi6.1670.0066.1486.187

E3 · formal · cited by C4, C5, C6, C7

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,*,*)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

  1. Impose E(1+ϵt)=1E(1 + \epsilon_t) = 1 on each distribution: μ=1\mu = 1 for Normal and Inverse Gaussian, μ=−σ2/2\mu = -\sigma^2 / 2 for Log-Normal, and ξ=σ−2\xi = \sigma^{-2} for Gamma. (moment_constraint; from S, e, c, t, i, o, n, , 4, ., 1, ,, , T, a, b, l, e, , 2)
  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)
  3. Differentiate log-likelihoods with respect to σ2\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)

E4 · formal · cited by C8

Analytical derivation of closed-form conditional hh-steps-ahead mean and variance, one-step-ahead predictive distributions via scalability, and smoothing parameter domain restrictions for ETS(M,N,N)ETS(M,N,N) (Section 4.2, Eqs. 4.2-4.7).

Foundation: classical_probability · Assumes: C12, C13

  1. Expand level recursion lt+h=lt∏j=1h(1+αϵt+j)l_{t+h} = l_t \prod_{j=1}^h (1 + \alpha \epsilon_{t+j}) under i.i.d. unit-mean errors to derive E(lt+h|t)=ltE(l_{t+h}|t) = l_t and V(lt+h|t)=lt2((1+α2σ2)h−1)V(l_{t+h}|t) = l_t^2((1 + \alpha^2 \sigma^2)^h - 1). (independent_product_moments; from S, e, c, t, i, o, n, , 4, ., 2, ,, , E, q, ., , (, 4, ., 5, ))
  2. Enforce positivity of level update 1+αϵt>01 + \alpha \epsilon_t > 0 for all 1+ϵt>01 + \epsilon_t > 0 to establish α∈[0,1]\alpha \in [0, 1]. (positivity_bound; from S, e, c, t, i, o, n, , 4, ., 2, ,, , E, q, ., , (, 4, ., 7, ))

E5 · simulation · cited by C9, C10

Monte Carlo simulation of 10,00010,000 series of length 500500 for each of five DGPs (ETS(M,Md,M)ETS(M,Md,M) with Normal, Inverse Gaussian, Gamma, Log-Normal, and Akram's exponentiated ETS(A,Ad,A)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 Normal error distribution, Inverse Gaussian error distribution, Gamma error distribution, Log-Normal error distribution, Akram's log-ETS model · cases Simulation experiment 2 synthetic series · procedure 48-step quantile forecast simulation procedure · lens scaled pinball score (SPS), quantile bias, quantile precision

E5.T1 · Mean scaled pinball scores (SPSSPS) across 10,00010,000 simulated series for applied models (rows) against data-generating processes (columns) over forecast horizon 4848.
Rowapplied_modelscaled pinball score (SPS) (DGP: Normal) (1)scaled pinball score (SPS) (DGP: Inverse Gaussian) (1)scaled pinball score (SPS) (DGP: Gamma) (1)scaled pinball score (SPS) (DGP: Log-Normal) (1)scaled pinball score (SPS) (DGP: Akram) (1)
r1Normal5.2555.2045.1939.4175.192
r2Inverse Gaussian5.35.2215.2249.4645.229
r3Gamma5.2875.1925.2319.3885.214
r4Log-Normal5.2155.2955.2249.4235.22
r5Akram5.115.0525.0749.185.089
r6Auto (AICc)5.2585.1755.2089.3325.178
E5.T2 · Quantile bias and quantile precision across 99 quantile levels (0.010.01 to 0.990.99) for applied models against data-generating processes.
Rowapplied_modelquantile bias (DGP: Normal) (1)quantile bias (DGP: Inverse Gaussian) (1)quantile bias (DGP: Gamma) (1)quantile bias (DGP: Log-Normal) (1)quantile bias (DGP: Akram) (1)quantile precision (DGP: Normal) (1)quantile precision (DGP: Inverse Gaussian) (1)quantile precision (DGP: Gamma) (1)quantile precision (DGP: Log-Normal) (1)quantile precision (DGP: Akram) (1)
r1Normal-0.0020.0030.0110.007-0.010.0070.0110.0050.0090.01
r2Inverse Gaussian-0.0040.0090.0010.001-0.0130.0090.0090.0060.0090.015
r3Gamma-0.0040.0080.0010.003-0.0120.0110.0090.0050.0080.014
r4Log-Normal-0.0050.00800.001-0.0130.0080.0040.0080.0040.014
r5Akram00.0040.010.006-0.0070.0040.010.0050.0080.007
r6Auto (AICc)-0.0030.00800.002-0.0120.0080.0040.0090.0070.013

E6 · quant · benchmark · secondary · n 511 · cited by C11

Empirical application of automatic pure multiplicative ETS selection via smooth::auto.adam() to 511511 daily EV charging observations (holdout 1414) and a low-volume initial 4949-observation subsample (holdout 1414) comparing Gamma and Normal error distributions (Section 6, Figs. 4-6).

Setup: studied ETS(M,Md,M), ETS(A,Ad,A), Gamma error distribution, Normal error distribution · cases Daily EV charging dataset · procedure two-dimensional AICc selection procedure · lens AICc

Limitations

L1 · model specification · reduces transferability · affects C1 · mitigation: none

Conditional expectations for pure multiplicative ETS models other than ETS(M,N,N)ETS(M,N,N) lack closed-form analytical expressions and require Monte Carlo simulation.

L2 · model specification · reduces transferability · affects C3 · mitigation: none

Simulation experiment 1 evaluates point forecast bias exclusively on ETS(M,Md,N)ETS(M,Md,N) with Inverse Gaussian errors and known true parameters.

L3 · model specification · reduces precision · affects C7 · 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 · reduces transferability · affects C9, C10 · mitigation: none

Simulation experiment 2 uses a single fixed parameter vector with error standard deviation σ=0.1\sigma = 0.1, where multiplicative errors remain small enough that Normal and logarithmic approximations behave regularly.

L5 · sampling · reduces transferability · affects C11 · 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,*)ETS(M,M,*)?

References

  1. Svetunkov and Boylan (2024) Svetunkov, I., & Boylan, J. E. (2024). Staying positive: challenges and solutions in using pure multiplicative ETS models. IMA Journal of Management Mathematics, 35(3), 403–425. doi:10.1093/imaman/dpad028 Used by: A0
  2. entrylib:2026.000003@v4#C7 entrylib:2026.000003@v4#C7 Used by: C7
  3. 978-3-540-71918-2 978-3-540-71918-2 Used by: C1, D2, D3, D7, D12, D13
  4. Akram et al. (2009) Akram, M., Hyndman, R. J., & Ord, J. K. (2009). EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA. Australian &amp; New Zealand Journal of Statistics, 51(4), 415–432. doi:10.1111/j.1467-842X.2009.00555.x Used by: C4, D10
  5. doi:10.1080/03610927808837599 doi:10.1080/03610927808837599 Used by: D9
  6. 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

Artefacts

  • A0 · manuscript · Svetunkov and Boylan (2024) The published article, IMA Journal of Management Mathematics (2023). dpad028
  • A1 · code R simulation and estimation code using smooth package v3.2.0 (shared on reasonable request to the corresponding author). smooth
  • A2 · data Daily electric vehicle charging demand dataset (511 observations) analysed in Section 6 (restricted by non-disclosure agreement). dpad028
  • A3 · table Full simulation results tables (Table 1 from Section 3 and Tables 4-5 from Section 5) exported as CSV. simulation-results.csv

Score

Support of the paper: 52, Weakly supported

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. Every issue as text, for an agent preparing the next version.

The mean of its 6 principal claims; the lowest is C9 at 31. Open problems in the principal claims and what they rest on, each counted once, with the most it costs a claim:

  1. −45 Checkable Verifiability: unverifiable. In C3, C9, C10, C11. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
  2. −25 Evidence Rests on an unchecked proof. In C1, C2, C3, C4, C5, C6, C7, C8, C9, C10. 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. −10 Open problems A principal claim that declares no limitation. In C4. To remove it: Declare what could bias, reverse or limit the claim within its scope, each as a limitation that affects it.
  4. −5 Plan A principal claim reported without a plan recorded beforehand. In C3, C4, C9, C10. 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.
  5. −5 Context A principal claim related to no earlier work. In C9. To remove it: Relate the claim to the earlier work it extends, addresses or departs from.
  6. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. In C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  7. −1 Open problems Limitation L1 (reduces transferability, not mitigated): Conditional expectations for pure multiplicative ETS models other than ETS(M,N,N)ETS(M,N,N) lack closed-form analytical expressions and require Monte Carlo simulation. In C1. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
  8. −1 Open problems Limitation L2 (reduces transferability, not mitigated): Simulation experiment 1 evaluates point forecast bias exclusively on ETS(M,Md,N)ETS(M,Md,N) with Inverse Gaussian errors and known true parameters. In C3. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
  9. −1 Open problems Limitation L4 (reduces transferability, not mitigated): Simulation experiment 2 uses a single fixed parameter vector with error standard deviation σ=0.1\sigma = 0.1, where multiplicative errors remain small enough that Normal and logarithmic approximations behave regularly. In C9, C10. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

Claims

C1: 76, Partly supported

Novelty: extends earlier work, by the authors

  1. −20 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.
  2. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  3. −1 Open problems Limitation L1 (reduces transferability, not mitigated): Conditional expectations for pure multiplicative ETS models other than ETS(M,N,N)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.

C2: 72, Partly supported

Novelty: extends earlier work, by the authors

  1. −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.
  2. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C3: 36, Weakly supported

Novelty: new, by the authors

  1. −45 Checkable Verifiability: unverifiable. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
  2. −10 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. −5 Plan 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.
  4. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  5. −1 Open problems Limitation L2 (reduces transferability, not mitigated): Simulation experiment 1 evaluates point forecast bias exclusively on ETS(M,Md,N)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.

C4: 62, Partly supported

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

  1. −20 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.
  2. −10 Open problems 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.
  3. −5 Plan 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.
  4. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C5: 72, Partly supported

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

  1. −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.
  2. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C6: 72, Partly supported

Novelty: new, by the authors

  1. −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.
  2. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C7: 74, Partly supported

Novelty: refutes earlier work, by the authors

  1. −20 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.
  2. −3 Open problems Limitation L3 (reduces precision, not mitigated): The Log-Normal concentrated log-likelihood under unit-mean restriction is non-convex, creating potential local optima during numerical optimisation. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
  3. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C8: 72, Partly supported

Novelty: extends earlier work, by the authors

  1. −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.
  2. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C9: 31, Weakly supported

Novelty: new, by the authors · in question: close match on EntryLib not acknowledged

  1. −45 Checkable Verifiability: unverifiable. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
  2. −10 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. −5 Plan 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.
  4. −5 Context A principal claim related to no earlier work. To remove it: Relate the claim to the earlier work it extends, addresses or departs from.
  5. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  6. −1 Open problems Limitation L4 (reduces transferability, not mitigated): Simulation experiment 2 uses a single fixed parameter vector with error standard deviation σ=0.1\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.

C10: 36, Weakly supported

Novelty: new, by the authors · in question: close match on EntryLib not acknowledged

  1. −45 Checkable Verifiability: unverifiable. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
  2. −10 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. −5 Plan 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.
  4. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  5. −1 Open problems Limitation L4 (reduces transferability, not mitigated): Simulation experiment 2 uses a single fixed parameter vector with error standard deviation σ=0.1\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.

C11: 46, Weakly supported

Novelty: new, by the authors

  1. −45 Checkable Verifiability: unverifiable. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
  2. −5 Context Numbers in the statement match no recorded result. To remove it: Record each number as a result or table cell of the evidence the claim cites, and list it in the support's results; or correct the number.
  3. −3 Open problems Assumption C12 is not checked: The multiplicative error term (1+ϵt)(1+\epsilon_t) in pure multiplicative ETS models is independent and identically distributed with unit expectation E(1+ϵt)=1E(1+\epsilon_t) = 1. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  4. −1 Open problems Limitation L5 (reduces transferability, not mitigated): The real-life demonstration relies on a single proprietary daily electric vehicle charging series that cannot be publicly shared under a non-disclosure agreement. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

Versions

  • Version 1 (this page) · 12 November 2026 at 21:53 UTC · active

Every version is kept in the public registry (github.com/entrylib/registry).