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Mathematics

11 principal claims

100C9 · descriptive · principal · novelty: new, by the authors
For iETS(M,N,N) in the conventional multiplicative ETS form, the normal-drift mixture density of the demand size after an interval had a largest Kullback-Leibler divergence per observation from the exact density at least three orders of magnitude smaller than that of the one-step density, for the Gamma, the log-normal and the inverse Gaussian distribution.
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
96C10 · formal · principal · novelty: new, by the authors
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 $p_t\hat\mu_t$ .
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
95C1 · formal · principal · novelty: new, by the authors
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 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.
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
95C2 · formal · principal · novelty: refutes earlier work, by the authors
The stationary points of the entropy-augmented likelihood of iETS differ in general from those of the observed-data likelihood: at a fixed point of the EM algorithm the gradient of the expected complete-data log-likelihood equals the score of the observed-data likelihood, while the term $-H_z(\theta)$ of each period with zero demand has a gradient that is not zero in general, and for normal sizes with fixed location the two objectives are stationary at different variances.
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
95C3 · formal · principal · novelty: new, by the authors
For normally distributed demand sizes with fixed location, the maximiser in the variance of the entropy-augmented likelihood is $\frac{T_1}{T}\hat\sigma^2$ , where $\hat\sigma^2=\frac{1}{T_1}\sum_{O_t=1}e_t^2$ is the fixed point of the EM algorithm, $T$ the number of periods and $T_1$ the number of periods with observed sizes.
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
92C4 · formal · principal · novelty: extends earlier work, by the authors
For a pure additive single source of error model with normal errors and unobserved periods, the 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 $\hat\sigma^2=\frac{1}{T_1}\sum_{O_t=1}\frac{v_t^2}{f_t}.$
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
92C5 · formal · principal · novelty: refutes earlier work, by the authors
For pure additive single source of error models with normal errors and unobserved periods, if $c_i=\mathbf w'\mathbf F^{i-1}\mathbf g\neq0$ for some $i$ shorter than an interval, the 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 $\sigma^2f_t>\sigma^2$ .
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
91C6 · formal · principal · novelty: new, by the authors
For the ADAM multiplicative ETS with the log-normal distribution and unobserved periods, the 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.
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
91C7 · formal · principal · novelty: new, by the authors
For the 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 $m=\frac{1}{T_1}\sum\log^2(1+\epsilon_t)$ .
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
91C8 · formal · principal · novelty: refutes earlier work, by the authors
The log-normal scale estimator $2(1-\sqrt{1-m})$ used in ADAM, with $m=\frac{1}{T_1}\sum\log^2(1+\epsilon_t)$ , is not the maximum likelihood estimator: it overestimates the scale $s$ by about 35% at $s=0.5$ (0.677 against 0.5 when $m=s+s^2/4$ ) and is undefined for $m>1$ .
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2
91C11 · methodological · principal · novelty: new, by the authors
The smooth package for R and Python implements the observed-data likelihood without the entropy term, the SSOE Kalman filter for the pure additive models with the normal distribution and in logarithms for the ADAM multiplicative ETS with the log-normal distribution, the 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.
The observed-data likelihood of ADAM and iETS with unobserved periods · entrylib:2026.000003@v2