CLAIM C11 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C11

DEFINITIONS
D2 [concept] unobserved period: A period $t$ at which the potential value $z_t$ of the modelled variable is not observed ($O_t=0$): a period with zero demand in an intermittent demand model, where the demand size is not observed, or a period with a missing value.
D3 [concept] observed-data likelihood: The joint density of the observed data only, obtained by integrating the complete-data likelihood over the values $\mathbf z_{\mathrm{mis}}$ at the unobserved periods: $$L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$$
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 $m$, $$\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\alpha\epsilon_t) \\ b_t &= b_{t-1}^{\phi}(1+\beta\epsilon_t) \\ s_t &= s_{t-m}(1+\gamma\epsilon_t). \end{aligned}$$
D8 [method] SSOE Kalman filter: The recursion for the conditional mean $\mathbf a_t$ and scaled covariance $\mathbf P_t$ of the state of a linear single source of error model: at an observed period $$\begin{aligned} v_t &= z_t-\mathbf w'\mathbf a_t \\ f_t &= \mathbf w'\mathbf P_t\mathbf w+1 \\ \mathbf k_t &= (\mathbf F\mathbf P_t\mathbf w+\mathbf g)/f_t \\ \mathbf a_{t+1} &= \mathbf F\mathbf a_t+\mathbf k_t v_t \\ \mathbf P_{t+1} &= \mathbf F\mathbf P_t\mathbf F'+\mathbf g\mathbf g'-\mathbf k_t\mathbf k_t'f_t; \end{aligned}$$ at an unobserved period $$\begin{aligned} \mathbf a_{t+1} &= \mathbf F\mathbf a_t \\ \mathbf P_{t+1} &= \mathbf F\mathbf P_t\mathbf F'+\mathbf g\mathbf g'; \end{aligned}$$ started from $\mathbf a_1=\mathbf x_0$, $\mathbf P_1=\mathbf 0$.
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.

CLAIMS
C11 [methodological; principal; support: demonstration artefacts A2; uses C1, C4, C6, C7; strength moderate (author); verifiability open]
  The smooth package for R and Python implements the [[D3:observed-data likelihood]] without the entropy term, the [[D8:SSOE Kalman filter]] for the pure additive models with the normal distribution and in logarithms for the [[D6:ADAM multiplicative ETS]] with the log-normal distribution, the [[D10:normal-drift mixture density]] with moment-equation scales for the other pure multiplicative models, and the corrected log-normal scale, in one C++ routine shared by both languages.
  scope: setting: smooth package for R and Python, master branch at commit 8a5fbbb3
  supersedes B4, at Appendix B, eq. B.6, 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.
C1 [formal; principal; support: formal_proof E1; assumes C30, C31, C32; verification agent_checked; verifiability open]
  For a model of intermittent demand $y_t=o_tz_t$ with Bernoulli occurrence $o_t\sim\text{Bernoulli}(p_t)$ independent of a positive continuous size $z_t$, the [[D3:observed-data likelihood]] is $$\sum_{O_t=1}\log f(z_t\mid Y_{t-1};\theta) + \sum_{O_t=1}\log p_t + \sum_{O_t=0}\log(1-p_t),$$ in which the unobserved sizes do not appear and the size and occurrence parts separate.
  scope: population: models of intermittent demand $y_t=o_tz_t$ (iETS, ADAM with an occurrence part) | data_type: intermittent demand time series | conditions: Bernoulli occurrence independent of a continuous demand size
  extends B1: Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood.
  extends B4, at Appendix B, eq. B.6, 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.
C4 [formal; principal; extension; support: formal_proof E4; assumes C30, C33, C34, C35; uses C1; verification agent_checked; verifiability open]
  For a pure additive single source of error model with normal errors and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] gives $z_t\mid Y_{t-1}\sim\mathcal N(\mathbf w'\mathbf a_t,\sigma^2 f_t)$ at the observed periods, the exact observed-data log-likelihood $$-\frac12\sum_{O_t=1}\left[\log(2\pi\sigma^2f_t)+\frac{v_t^2}{\sigma^2f_t}\right],$$ and the concentrated variance $$\hat\sigma^2=\frac{1}{T_1}\sum_{O_t=1}\frac{v_t^2}{f_t}.$$
  scope: population: pure additive single source of error models (ETS with additive components, ARIMA, regression) | data_type: univariate time series with unobserved periods | conditions: normally distributed errors; initial states treated as fixed parameters | excludes: models mixing additive and multiplicative components
  extends B6, at Kalman filter for state space models with correlated disturbances: The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated.
  extends B7, at Kalman filter for state space models with correlated disturbances: The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations.
C6 [formal; principal; support: formal_proof E9; assumes C30, C33, C34, C36; uses C4, C12; verification agent_checked; verifiability open]
  For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] applied to $\log z_t$, with the known error mean $-s/2$ added to the prediction and, multiplied by $\mathbf g$, to the propagation, gives the exact observed-data likelihood, and the maximum likelihood scale $s$ is the positive root of $$\Big(\sum \frac{b_t^2}{f_t}\Big)s^2+T_1s-\sum\frac{(v_t^0)^2}{f_t}=0,$$ where $v_t(s)=v_t^0+sb_t$ is the filter's prediction error.
  scope: population: pure multiplicative ETS in the ADAM form | data_type: positive univariate time series with unobserved periods | conditions: log-normal distribution with $\E(1+\epsilon_t)=1$
  extends B5, own earlier work: The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed.
C7 [formal; principal; support: formal_proof E10; assumes C30, C36; uses C6; verification agent_checked; verifiability open]
  For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and all periods observed, the maximum likelihood estimate of the scale is $\hat s=2(\sqrt{1+m}-1)$ with $m=\frac{1}{T_1}\sum\log^2(1+\epsilon_t)$.
  scope: population: pure multiplicative ETS in the ADAM form | data_type: positive univariate time series | conditions: log-normal distribution with $\E(1+\epsilon_t)=1$; every period observed
  extends B5, own earlier work: The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed.
C12 [formal; supporting; support: formal_proof E8; verification agent_checked; verifiability open]
  In the [[D6:ADAM multiplicative ETS]], the logarithms of the observation and of the components follow a linear single source of error model with innovation $u_t=\log(1+\epsilon_t)$ and the measurement, transition and persistence of the corresponding additive model, whereas in the [[D7:conventional multiplicative ETS]] the innovation of component $k$ is $\log(1+g_k\epsilon_t)$, which is not proportional to $u_t$.
  scope: population: pure multiplicative ETS in the ADAM and conventional forms
C30 [descriptive; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  The errors $\epsilon_t$ of the single source of error model are independent across periods.
C31 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$.
C32 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$.
C33 [descriptive; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  Whether a period is observed carries no information about the unobserved value $z_t$ beyond the model, so that the missing values are missing at random.
C34 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The initial states of the single source of error model are fixed parameters, so that the state covariance is zero at the start of the sample.
C35 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The errors of the pure additive model are normally distributed with constant variance $\sigma^2$.
C36 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$, so that $\E(1+\epsilon_t)=1$.

EVIDENCE
E1 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E1.1 $P(y_t=0\mid Y_{t-1})=(1-p_t)\int f(z\mid Y_{t-1})dz=1-p_t$ (by C31, C32; independence and continuity)
  E1.2 $y_t>0$ has density $p_tf(y_t\mid Y_{t-1})$ (by C31; independence)
  E1.3 The joint density of the observed data is the product of these conditional terms (by E1.1, E1.2; prediction-error decomposition)
E4 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C30, C33, C34, C35
  E4.1 $(\mathbf x_{t-1},z_t,\mathbf x_t)$ is jointly Gaussian given $Y_{t-1}$ (by C35, C30; linearity of Gaussian models)
  E4.2 $$\begin{aligned} \E(z_t\mid Y_{t-1}) &= \mathbf w'\mathbf a_t \\ \Var(z_t\mid Y_{t-1}) &= \sigma^2f_t \\ \Cov(\mathbf x_t,z_t\mid Y_{t-1}) &= \sigma^2(\mathbf F\mathbf P_t\mathbf w+\mathbf g) \end{aligned}$$ (by E4.1; moments of linear combinations)
  E4.3 Conditioning on $z_t$ gives the update; unobserved periods only propagate (by E4.2, C33; Gaussian conditioning)
  E4.4 The likelihood follows from the observed-data decomposition, and $\sigma^2$ concentrates out because $f_t$, $\mathbf k_t$ do not depend on it (by E4.3, C1; prediction-error decomposition)
E8 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E8.1 $\log z_t=\log l_{t-1}+\phi\log b_{t-1}+\log s_{t-m}+u_t$ (by D6; logarithm)
  E8.2 $\log l_t=\log l_{t-1}+\phi\log b_{t-1}+\alpha u_t$, $\log b_t=\phi\log b_{t-1}+\beta u_t$ in the ADAM form (by D6; logarithm)
  E8.3 $\log l_t=\log l_{t-1}+\phi\log b_{t-1}+\log(1+\alpha\epsilon_t)$ in the conventional form (by D7; logarithm)
E9 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C30, C33, C34, C36
  E9.1 $\log z_t$ follows a linear Gaussian SSOE model with known error mean $-s/2$ (by C12, C36; linearity in logarithms)
  E9.2 The filter of the linear Gaussian model with the Jacobian $-\sum\log z_t$ gives the exact likelihood (by C33, C34, C4, E9.1; change of variables)
  E9.3 Predictions are linear in $-s/2$ with gain-only coefficients, so $v_t(s)=v_t^0+sb_t$ (by E9.2; linearity)
  E9.4 The score in $s$ times $2s^2$ gives $(\sum b_t^2/f_t)s^2+T_1s-\sum(v_t^0)^2/f_t=0$ (by E9.3; differentiation)
E10 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C36
  E10.1 $f_t=1$, $b_t=1/2$ give $s^2+4s-4m=0$ (by C6, C15; substitution)
  E10.2 The positive root is $2(\sqrt{1+m}-1)$; $2(1-\sqrt{1-m})$ solves $s^2-4s+4m=0$ instead (by E10.1; quadratic formula)
  E10.3 At $s=0.5$, $\E m=s+s^2/4=0.5625$ and $2(1-\sqrt{1-0.5625})=0.677$ (by E10.2, C36; arithmetic)

ARTEFACTS
A1 [manuscript] LaTeX source of the draft 'Estimating exponential smoothing with unobserved periods: the observed-data likelihood of ADAM and iETS' (October 2026).
A2 [software] smooth package for R and Python, including the SSOE Kalman filter in the shared C++ core (src/headers/adamCore.h).
A10 [proof] Report of the first round of the independent proof check by the contributor P6 (2026-11-11): for C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19, the re-derivation, the numerical spot checks, the outcome and the gaps found.
A11 [proof] Report of the second round of the independent proof check by P6 (2026-11-11), on C2, C5, C10, C14 and C19 as corrected after the first round, and on the smaller changes to C1, C6, C8 and C16.
A14 [code] Numerical spot checks of the first round of the proof check (Python with numpy); the report A10 says which claim each serves.
A16 [code] Numerical spot checks of the proof check (Python with numpy), used in both rounds; the reports A10 and A11 say which claim each serves.

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C11, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C11. https://entrylib.org/p/2026.000003/C11
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