CLAIM C10 of entrylib:2026.000006 Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
licence CC-BY-4.0; contributions C10

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]
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]
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 $48$.
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.

CLAIMS
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 $1 + \epsilon_t > 0$.

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)
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 |

LIMITATIONS
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.

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

REFERENCES
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 C10, in: Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1#C10. https://entrylib.org/p/2026.000006/C10
The claimset it is from: Svetunkov, I. (guarantor) (2026). Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1. https://entrylib.org/p/2026.000006@v1