The claim
The log-normal scale estimator used in ADAM, with , is not the maximum likelihood estimator: it overestimates the scale by about 35% at (0.677 against 0.5 when ) and is undefined for .
Scope: population: pure multiplicative ETS in the ADAM form; data: positive univariate time series; conditions: log-normal distribution with , every period observed · Proof: E10 · Uses: C15, C7 · Assumes: C36 · Contradicts: B5, estimator of the scale of the log-normal distribution · Verification: checked by an agent
For a model of intermittent demand with Bernoulli occurrence independent of a positive continuous size , the observed-data likelihood is in which the unobserved sizes do not appear and the size and occurrence parts separate.
Scope: population: models of intermittent demand (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
For a pure additive single source of error model with normal errors and unobserved periods, the SSOE Kalman filter gives at the observed periods, the exact observed-data log-likelihood and the concentrated variance
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
For the ADAM multiplicative ETS with the log-normal distribution and unobserved periods, the SSOE Kalman filter applied to , with the known error mean added to the prediction and, multiplied by , to the propagation, gives the exact observed-data likelihood, and the maximum likelihood scale is the positive root of where is the filter's prediction error.
Scope: population: pure multiplicative ETS in the ADAM form; data: positive univariate time series with unobserved periods; conditions: log-normal distribution with · Proof: E9 · Uses: C4, C12 · Assumes: C30, C33, C34, C36 · Extends: B5 · Verification: checked by an agent
For the ADAM multiplicative ETS with the log-normal distribution and all periods observed, the maximum likelihood estimate of the scale is with .
Scope: population: pure multiplicative ETS in the ADAM form; data: positive univariate time series; conditions: log-normal distribution with , every period observed · Proof: E10 · Uses: C6 · Assumes: C30, C36 · Extends: B5 · Verification: checked by an agent
Claims it rests on
In the ADAM multiplicative ETS, the logarithms of the observation and of the components follow a linear single source of error model with innovation and the measurement, transition and persistence of the corresponding additive model, whereas in the conventional multiplicative ETS the innovation of component is , which is not proportional to .
Scope: population: pure multiplicative ETS in the ADAM and conventional forms · Proof: E8 · Verification: checked by an agent · Used by: C6
When every period is observed, the SSOE Kalman filter keeps , has gain and , and coincides with the single source of error recursion and the likelihood of the one-step errors.
Scope: population: pure additive single source of error models; data: univariate time series; conditions: every period observed · Proof: E5 · Cited: Ord et al. (1997) · Cited: Snyder (1985) · Uses: C4 · Verification: checked by an agent · Used by: C8
Evidence
Analytical derivation in the source draft (artefact A1).
Foundation: Kolmogorov probability axioms · Assumes: C31, C32
Analytical derivation in the source draft (artefact A1).
Foundation: Kolmogorov probability axioms · Assumes: C30, C33, C34, C35
- is jointly Gaussian given (linearity of Gaussian models; from C35, C30)
- (moments of linear combinations; from E4.1)
- Conditioning on gives the update; unobserved periods only propagate (Gaussian conditioning; from E4.2, C33)
- The likelihood follows from the observed-data decomposition, and concentrates out because , do not depend on it (prediction-error decomposition; from E4.3, C1)
Analytical derivation in the source draft (artefact A1).
Foundation: Kolmogorov probability axioms
- With , and (substitution; from D8)
- (substitution; from E5.1)
- By induction from the filter is the SSOE recursion (induction; from E5.2)
Analytical derivation in the source draft (artefact A1).
Foundation: Kolmogorov probability axioms
Analytical derivation in the source draft (artefact A1).
Foundation: Kolmogorov probability axioms · Assumes: C30, C33, C34, C36
- follows a linear Gaussian SSOE model with known error mean (linearity in logarithms; from C12, C36)
- The filter of the linear Gaussian model with the Jacobian gives the exact likelihood (change of variables; from C33, C34, C4, E9.1)
- Predictions are linear in with gain-only coefficients, so (linearity; from E9.2)
- The score in times gives (differentiation; from E9.3)
Assumptions
The errors of the single source of error model are independent across periods. Rested on by C1, C4, C6, C7, E4, E9.
In the intermittent demand model the occurrence is independent of the potential demand size . Rested on by C1, E1.
Whether a period is observed carries no information about the unobserved value beyond the model, so that the missing values are missing at random. Rested on by C4, C6, E4, E9.
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, C6, E4, E9.
Definitions
Methods 3
D6 ADAM multiplicative ETS: Pure multiplicative exponential smoothing in the form of the Augmented Dynamic Adaptive Model, in which each component is updated by the factor ; for ETS(M,Md,M) with seasonal period ,
D7 conventional multiplicative ETS: Pure multiplicative exponential smoothing in the form of Hyndman et al. (2008), in which each component is updated by the factor ; for ETS(M,Md,M) with seasonal period ,
D8 SSOE Kalman filter: The recursion for the conditional mean and scaled covariance of the state of a linear single source of error model: at an observed period at an unobserved period started from , .
Concepts 2
D2 unobserved period: A period at which the potential value of the modelled variable is not observed (): 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 at the unobserved periods:
Context
- Contradicts B5, estimator of the scale of the log-normal distribution · own earlier work · from C8
- 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.
- 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.
- 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 B5 · own earlier work · from C6
- 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.
- Extends B5 · own earlier work · from C7
- 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.
References
- Ord et al. (1997) Ord, J. K., Koehler, A. B., & Snyder, R. D. (1997). Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models. Journal of the American Statistical Association, 92(440), 1621–1629. doi:10.1080/01621459.1997.10473684 Used by: C15
- Snyder (1985) Snyder, R. D. (1985). Recursive Estimation of Dynamic Linear Models. Journal of the Royal Statistical Society Series B: Statistical Methodology, 47(2), 272–276. doi:10.1111/j.2517-6161.1985.tb01355.x Used by: C15
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.
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.
C8: 91, High
Novelty: refutes earlier work, by the authors · no close match on EntryLib
- −3 Open problems Assumption C36 is not checked: In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$, so that $\E(1+\epsilon_t)=1$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
- −1 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.
- −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.
- −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.
- −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.
- −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.
- −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
- −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.
- −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.
- −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.
C4: 92, High
Novelty: extends earlier work, by the authors · no close match on EntryLib
- −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.
- −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.
- −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.
- −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.
- −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.
- −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.
C6: 91, High
Novelty: new, by the authors · no close match on EntryLib
- −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.
- −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.
- −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.
- −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.
- −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.
- −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.
- −1 Open problems Assumption C36 is not checked: In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$, so that $\E(1+\epsilon_t)=1$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
C7: 91, High
Novelty: new, by the authors · no close match on EntryLib
- −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.
- −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.
- −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.
- −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.
- −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.
- −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.
- −1 Open problems Assumption C36 is not checked: In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$, so that $\E(1+\epsilon_t)=1$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
C15: 92, High
Novelty: new, by the authors
- −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.
- −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.
- −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.
- −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.
- −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.
- −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.