CLAIM C5 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C5

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: $$L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$$
D5 [method] zero-error recursion: The estimator used in ADAM and in iETS (Svetunkov and Boylan 2023, Appendix C): the single source of error recursion run with $\epsilon_t=0$ at the unobserved periods and with the fixed persistence vector $\mathbf g$ at the observed ones, with the likelihood taken as the product of one-step densities at the observed periods. [source B4]
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 $\mathbf a_1=\mathbf x_0$, $\mathbf P_1=\mathbf 0$.

CLAIMS
C5 [formal; principal; refutation; support: formal_proof E6; assumes C30, C33, C34, C35; uses C4; verification agent_checked; verifiability open]
  For pure additive single source of error models with normal errors and [[D2:unobserved period]]s, if $c_i=\mathbf w'\mathbf F^{i-1}\mathbf g\neq0$ for some $i$ shorter than an interval, the [[D5:zero-error recursion]] likelihood is not the observed-data likelihood and its maximiser in the error variance is in general not the maximum likelihood estimate, because at the observed period after such an interval the recursion uses the error variance $\sigma^2$ instead of $\sigma^2f_t>\sigma^2$.
  scope: population: pure additive single source of error models | data_type: univariate time series with unobserved periods | conditions: normally distributed errors; $\mathbf w'\mathbf F^{i-1}\mathbf g\neq0$ for some $i$ shorter than an interval
  contradicts B4, at Appendix C (estimation with zero errors at the unobserved periods), 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.
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 $\sigma^2$.

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)
E6 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E6.1 After $j-1$ unobserved periods from $\mathbf P=\mathbf 0$, $f=1+\sum_{i<j}c_i^2$ with $c_i=\mathbf w'\mathbf F^{i-1}\mathbf g$ (by D8; iteration of the propagation)
  E6.2 $f_t>1$ when $c_i\neq0$ for some $i<j$ (by E6.1; substitution)
  E6.3 The zero-error recursion uses variance $\sigma^2$ at every observed period, so its objective differs from the exact likelihood (by E6.2, D5, C4; comparison)
  E6.4 Both predict the first observed period after the first interval from the states propagated by $\mathbf F$ with zero errors, so its prediction error $e_t$ is the same in both (by D5, D8; zero mean of the errors in the interval)
  E6.5 For given other parameters the maximiser in $\sigma^2$ is $\frac{1}{T_1}\sum e_t^2$ for the zero-error recursion and $\frac{1}{T_1}\sum v_t^2/f_t$ for the exact likelihood, which differ unless the data satisfy $\sum e_t^2=\sum v_t^2/f_t$, a term with $f_t>1$ and $v_t=e_t$ being present (by E6.2, E6.4, C4; differentiation in $\sigma^2$)

ARTEFACTS
A1 [manuscript] LaTeX source of the draft 'Estimating exponential smoothing with unobserved periods: the observed-data likelihood of ADAM and iETS' (October 2026).
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.
A17 [code] Numerical spot checks of the second round of the proof check (Python with numpy); the report A11 says which claim each serves.

REFERENCES
B4 10.1016/j.ijpe.2023.109013 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C5, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C5. https://entrylib.org/p/2026.000003/C5
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