CLAIM C9 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C9

DEFINITIONS
D6 [method] ADAM multiplicative ETS: Pure multiplicative exponential smoothing in the form of the Augmented Dynamic Adaptive Model, in which each component $k$ is updated by the factor $(1+\epsilon_t)^{g_k}$; for ETS(M,Md,M) with seasonal period $m$, $$\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\epsilon_t)^{\alpha} \\ b_t &= b_{t-1}^{\phi}(1+\epsilon_t)^{\beta} \\ s_t &= s_{t-m}(1+\epsilon_t)^{\gamma}. \end{aligned}$$
D7 [method] conventional multiplicative ETS: Pure multiplicative exponential smoothing in the form of Hyndman et al. (2008), in which each component $k$ is updated by the factor $1+g_k\epsilon_t$; for ETS(M,Md,M) with seasonal period $m$, $$\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\alpha\epsilon_t) \\ b_t &= b_{t-1}^{\phi}(1+\beta\epsilon_t) \\ s_t &= s_{t-m}(1+\gamma\epsilon_t). \end{aligned}$$
D9 [concept] drift of the states: The change $d_t=\log(\mu_t/\hat\mu_t)$ of the logarithm of the location of a pure multiplicative model accumulated over the unobserved periods since the last observed period, relative to the point forecast $\hat\mu_t$ with zero errors.
D10 [method] normal-drift mixture density: Approximate predictive density of a demand size after $j_t-1$ unobserved periods: the one-step density $f_z(z_t\mid\hat\mu_t e^{d})$ mixed over a normal [[D9:drift of the states]] $d\sim\mathcal N(m_t,w_t)$, with $m_t$, $w_t$ computed by 32-node Gauss-Legendre quadrature on the quantile function of $1+\epsilon$ and the mixture by Gauss-Hermite quadrature.
D11 [method] one-step density: The density of a demand size after an interval taken as the one-step density $f_z(z_t\mid\hat\mu_t)$ of the model, without the [[D9:drift of the states]] over the unobserved periods.
D12 [data] iETS(M,N,N): Intermittent demand sizes $$z_t=l_{t-1}(1+\epsilon_t)$$ with $\E(1+\epsilon_t)=1$ and $\Var(1+\epsilon_t)=s$, Gamma, log-normal or inverse Gaussian $1+\epsilon_t$ (the greybox parameterisation of the package) and the level moving at every period, observed or not: $$l_t=l_{t-1}(1+\epsilon_t)^\alpha$$ in the [[D6:ADAM multiplicative ETS]] and $$l_t=l_{t-1}(1+\alpha\epsilon_t)$$ in the [[D7:conventional multiplicative ETS]].
D13 [design] exact density by convolution: The exact density of the size after an interval, computed on a grid of step 0.001 in $\log z$ by FFT convolution with tails cut at 1e-12, compared with an approximation by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20 (kl_mixture.py); a refinement of 164 cells halved the step and cut the tails at 1e-14.
D14 [variable] Kullback-Leibler divergence: The divergence $\int g\log(g/h)$ of an approximate density $h$ from the exact density $g$, per observation; over a grid, its largest value.

CLAIMS
C9 [descriptive; principal; support: simulation E20 (E20.T1.r1, E20.T1.r2, E20.T1.r3) artefacts A3, A4, A5; assumes C37; strength moderate (author); verifiability open]
  For [[D12:iETS(M,N,N)]] in the [[D7:conventional multiplicative ETS]] form, the [[D10:normal-drift mixture density]] of the demand size after an interval had a largest [[D14:Kullback-Leibler divergence]] per observation from the exact density at least three orders of magnitude smaller than that of [[D11:the one-step density]], for the Gamma, the log-normal and the inverse Gaussian distribution.
  scope: population: iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes | conditions: scale $s\in[0.05,1]$; smoothing parameter $\alpha\in[0.1,0.6]$; interval length $j\in[2,20]$; states known at the previous observed period | excludes: models with trend or seasonality; ADAM multiplicative ETS
  extends B4, own earlier work: Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand.
C37 [descriptive; supporting; assumption (methodological, tested); verifiability not_applicable]
  The [[D9:drift of the states]] of a pure multiplicative model over an interval is approximately normally distributed, with the states known at the previous observed period.

EVIDENCE
E20 [simulation; data_origin primary; runs 1] Comparison, for iETS(M,N,N) under the ADAM and conventional multiplicative ETS forms, of the normal-drift mixture density (drift moments by 32-node Gauss-Legendre, mixture by 16-node Gauss-Hermite) and of the one-step density against the exact density of the size after an interval, computed on a grid of step 0.001 in log z by FFT convolution with tails cut at 1e-12, by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20; a 164-cell refinement run halved the step and cut tails at 1e-14. The run recorded smooth commit f9d062ee with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.
  setup: studied D10, D11; cases D12, D6, D7; procedure D13; lens D14
  E20.T1 Largest Kullback-Leibler divergence per observation from the exact density of the size after an interval, of the normal-drift mixture density and of the one-step density, for iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes in the conventional and the ADAM form {model: iETS(M,N,N)}
  | row | distribution | ets_form | largest KL divergence per observation, normal-drift mixture vs exact density | largest KL divergence per observation, one-step vs exact density |
  | r1 | dgamma | conventional | 0.00114767 | 2.35934 |
  | r2 | dlnorm | conventional | 0.00447805 | 5.94732 |
  | r3 | dinvgauss | conventional | 0.00399623 | 47.5452 |
  | r4 | dgamma | adam | 0.0145197 | 4.58358 |
  | r5 | dlnorm | adam | 9.63871e-07 | 18.6354 |
  | r6 | dinvgauss | adam | 0.000446034 | 630.772 |
  outputs: largest KL per observation, conventional form: mixture 0.00115 (Gamma), 0.00448 (log-normal), 0.00400 (inverse Gaussian), one-step 2.359, 5.947, 47.545; ADAM form: mixture 0.01452, 0 (closed form), 0.00045, one-step 4.584, 18.635, 630.772
  seeds: [20261015]

ARTEFACTS
A3 [code] Python scripts (kl_mixture.py, scale_recovery.py, interval_bounds.py, lowrank_timing.py with build_lowrank_variants.sh, kl_laplace.py) that produce the numerical results on the observed-data likelihood of ADAM and iETS.
A4 [table] Kullback-Leibler divergence per observation of the normal-drift mixture density and of the one-step density from the exact density of the iETS(M,N,N) demand size after an interval, for 25,080 cells of distribution, ETS form, scale, smoothing parameter and interval length.
A5 [table] Refinement check of 164 cells of the iETS(M,N,N) Kullback-Leibler grid with half the grid step and a tail cut of 1e-14, with differences from the base run.

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C9, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C9. https://entrylib.org/p/2026.000003/C9
The claimset it is from: Svetunkov, I. (guarantor) (2026). The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2. https://entrylib.org/p/2026.000003@v2