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Claim C4 · formal · entrylib:2026.000003@v2 · 2026-11-11

Corrected · 11 November 2026 at 11:22 UTC

The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\mathbf w'\mathbf F^{i-1}\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed.

By Ivan Svetunkov (ORCID 0000-0001-7826-0281)

The observed-data likelihood of ADAM and iETS with unobserved periods · C4

Principal claim C4 of this claimset (Svetunkov, I. (guarantor), 2026), with what it rests on. Read the whole claimset.

Copies a message for ChatGPT, Claude or Gemini: paste it there.

The message

Read https://entrylib.org/p/2026.000003/C4, one claim of a research paper with the evidence it rests on. Tell me briefly what it says and how well it is supported. If you cannot open the link, tell me instead of guessing. Then wait for my questions.

The claim

92C4 · formal · extends earlier work · novelty: extends earlier work, by the authors · no close match on EntryLib

For a pure additive single source of error model with normal errors and unobserved periods, the SSOE Kalman filter gives zt∣Yt−1∼𝒩(𝐰′𝐚t,σ2ft)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 −12∑Ot=1[log⁡(2πσ2ft)+vt2σ2ft],-\frac12\sum_{O_t=1}\left[\log(2\pi\sigma^2f_t)+\frac{v_t^2}{\sigma^2f_t}\right], and the concentrated variance σ^2=1T1∑Ot=1vt2ft.\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: univariate time series with unobserved periods; conditions: normally distributed errors, initial states treated as fixed parameters; not for: models mixing additive and multiplicative components · Proof: E4 · Uses: C1 · Assumes: C30, C33, C34, C35 · Extends: B6, Kalman filter for state space models with correlated disturbances · Extends: B7, Kalman filter for state space models with correlated disturbances · Verification: checked by an agent

95C1 · formal · novelty: new, by the authors · no close match on EntryLib

For a model of intermittent demand yt=otzty_t=o_tz_t with Bernoulli occurrence ot∼Bernoulli(pt)o_t\sim\text{Bernoulli}(p_t) independent of a positive continuous size ztz_t, the observed-data likelihood is ∑Ot=1log⁡f(zt∣Yt−1;θ)+∑Ot=1log⁡pt+∑Ot=0log⁡(1−pt),\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 yt=otzty_t=o_tz_t (iETS, ADAM with an occurrence part); data: intermittent demand time series; conditions: Bernoulli occurrence independent of a continuous demand size · Proof: E1 · Assumes: C30, C31, C32 · Extends: B1 · Extends: B4, Appendix B, eq. B.6 · Verification: checked by an agent

Evidence

E1 · formal · cited by C1

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C31, C32

  1. P(yt=0∣Yt−1)=(1−pt)∫f(z∣Yt−1)dz=1−ptP(y_t=0\mid Y_{t-1})=(1-p_t)\int f(z\mid Y_{t-1})dz=1-p_t (independence and continuity; from C31, C32)
  2. yt>0y_t>0 has density ptf(yt∣Yt−1)p_tf(y_t\mid Y_{t-1}) (independence; from C31)
  3. The joint density of the observed data is the product of these conditional terms (prediction-error decomposition; from E1.1, E1.2)

E4 · formal · cited by C4

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C30, C33, C34, C35

  1. (𝐱t−1,zt,𝐱t)(\mathbf x_{t-1},z_t,\mathbf x_t) is jointly Gaussian given Yt−1Y_{t-1} (linearity of Gaussian models; from C35, C30)
  2. E⁡(zt∣Yt−1)=𝐰′𝐚tVar⁡(zt∣Yt−1)=σ2ftCov⁡(𝐱t,zt∣Yt−1)=σ2(𝐅𝐏t𝐰+𝐠)\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} (moments of linear combinations; from E4.1)
  3. Conditioning on ztz_t gives the update; unobserved periods only propagate (Gaussian conditioning; from E4.2, C33)
  4. The likelihood follows from the observed-data decomposition, and σ2\sigma^2 concentrates out because ftf_t, 𝐤t\mathbf k_t do not depend on it (prediction-error decomposition; from E4.3, C1)

Assumptions

C30 · statistical · unchecked

The errors ϵt\epsilon_t of the single source of error model are independent across periods. Rested on by C4, C1, E4.

C31 · theorem_condition · unchecked

In the intermittent demand model the occurrence oto_t is independent of the potential demand size ztz_t. Rested on by C1, E1.

C32 · theorem_condition · unchecked

The potential demand size ztz_t has a continuous distribution, so that P(zt=0)=0P(z_t=0)=0. Rested on by C1, E1.

C33 · statistical · unchecked

Whether a period is observed carries no information about the unobserved value ztz_t beyond the model, so that the missing values are missing at random. Rested on by C4, E4.

C34 · theorem_condition · unchecked

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. Rested on by C4, E4.

C35 · theorem_condition · unchecked

The errors of the pure additive model are normally distributed with constant variance σ2\sigma^2. Rested on by C4, E4.

Definitions

Methods 1

D8 SSOE Kalman filter: The recursion for the conditional mean 𝐚t\mathbf a_t and scaled covariance 𝐏t\mathbf P_t of the state of a linear single source of error model: at an observed period vt=zt−𝐰′𝐚tft=𝐰′𝐏t𝐰+1𝐤t=(𝐅𝐏t𝐰+𝐠)/ft𝐚t+1=𝐅𝐚t+𝐤tvt𝐏t+1=𝐅𝐏t𝐅′+𝐠𝐠′−𝐤t𝐤t′ft;\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 𝐚t+1=𝐅𝐚t𝐏t+1=𝐅𝐏t𝐅′+𝐠𝐠′;\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 𝐚1=𝐱0\mathbf a_1=\mathbf x_0, 𝐏1=𝟎\mathbf P_1=\mathbf 0.

Concepts 2

D2 unobserved period: A period tt at which the potential value ztz_t of the modelled variable is not observed (Ot=0O_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 observed-data likelihood: The joint density of the observed data only, obtained by integrating the complete-data likelihood over the values 𝐳mis\mathbf z_{\mathrm{mis}} at the unobserved periods: L(θ)=∫f(𝐲obs,𝐳mis;θ)d𝐳mis.L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.

Context

Extends B6, Kalman filter for state space models with correlated disturbances · from C4
The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated.
Extends B7, Kalman filter for state space models with correlated disturbances · from C4
The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations.
Extends B1 · from C1
Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood.
Extends B4, Appendix B, eq. B.6 · own earlier work · from C1
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.

Artefacts

Score

Scores are computed by rules from the claimset file (EntryLib rubric 0.2): they check the record, not the science, and may change as the rubric and reviews develop. The rubric.

C4: 92, High

Novelty: extends earlier work, by the authors · no close match on EntryLib

  1. −3 Open problems Assumption C30 is not checked: The errors $\epsilon_t$ of the single source of error model are independent across periods. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  2. −1 Open problems Assumption C31 is not checked: In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  3. −1 Open problems Assumption C32 is not checked: The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  4. −1 Open problems Assumption C33 is not checked: 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. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  5. −1 Open problems Assumption C34 is not checked: 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. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  6. −1 Open problems Assumption C35 is not checked: The errors of the pure additive model are normally distributed with constant variance $\sigma^2$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C1: 95, High

Novelty: new, by the authors · no close match on EntryLib

  1. −3 Open problems Assumption C30 is not checked: The errors $\epsilon_t$ of the single source of error model are independent across periods. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  2. −1 Open problems Assumption C31 is not checked: In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  3. −1 Open problems Assumption C32 is not checked: The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.