Pairwise Difference Representations of Central Moments: Skewness, Kurtosis, and Higher-order Recursions
Bibliographic record
Abstract
We develop pairwise-difference (Gini-type) representations of higher-order central moments for both general random variables and empirical moments. These representations eliminate the need for any location parameter. For third and fourth central moments, this yields pairwise-difference representations of skewness and kurtosis coefficients. We further derive a recursion that constructs higher-order representations from lower-order components, showing that all finite central moments possess such representations without reference to the mean. This is achieved by considering i.i.d. replications of the random variables, by interpreting central moments as covariances between a random variable and its powers, and by establishing recursions which link the pairwise-difference representation of any moment to lower order ones. Numerical summation identities are deduced. Finally, we use these identities to derive unbiased estimators of central moments and introduce a Monte Carlo approximation estimator that remains unbiased while greatly reducing the computation cost. Simulation results comparing the proposed estimators with the usual sample-moment (“natural”) estimators demonstrate substantial bias reduction across moment orders, with the largest improvements observed at higher orders
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.028 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".