CLAIM C10 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C10

CLAIMS
C10 [formal; principal; support: formal_proof E12; assumes C31, C32; verification agent_checked; verifiability open]
  For a model with an occurrence part, occurrence probability $p_t$ and positive sizes, a prediction interval $[0,u]$ of the demand with coverage $c>1-p_t$ that counts the zeros as covered has as upper bound $u$ the quantile at level $(c-(1-p_t))/p_t$ of the distribution of the demand size, whose mean, the conditional mean of the size, is the size forecast $\hat\mu_t$, not the demand forecast $p_t\hat\mu_t$.
  scope: population: models with an occurrence part (iETS, ADAM) | conditions: prediction interval of the demand of the form $[0,u]$ that counts zeros as covered; coverage $c>1-p_t$; Bernoulli occurrence independent of a positive continuous demand size
  extends B4, own earlier work: Svetunkov and Boylan forecast intermittent demand as the product of the occurrence probability and the forecast demand size.
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 $P(z_t=0)=0$.

EVIDENCE
E12 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E12.1 $$P(0\le y_t\le u)=(1-p_t)+p_tF_z(u)$$ for positive sizes and occurrence independent of the size (by C31, C32; law of total probability)
  E12.2 Setting it to $c$ with $c>1-p_t$ gives $F_z(u)=(c-(1-p_t))/p_t\in(0,1]$, so $u$ is the quantile of the size at that level (by E12.1; algebra)
  E12.3 The size has mean $\hat\mu_t$, the demand $\E(o_tz_t)=p_t\hat\mu_t$ (by C31; independence)

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.
A17 [code] Numerical spot checks of the second round of the proof check (Python with numpy); the report A11 says which claim each serves.

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C10, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C10. https://entrylib.org/p/2026.000003/C10
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