The empirical identity process: Asymptotics and applications
Bibliographic record
Abstract
Abstract When sampling independent observations drawn from the uniform distribution on the unit interval, as the sample size gets large the asymptotic behaviour of both the empirical distribution function and empirical quantile function is well known. In this article we study analogous asymptotic results for the function that is obtained by composing the empirical quantile function with the empirical distribution function. Since the former is the generalized inverse of the latter, the result will approximate the identity function. We define a scaled and centered version of this function—the empirical identity process —and prove it converges to a highly irregular limit process whose trajectories are not right‐continuous and impossible to study using standard probability in metric spaces. However, when this process is integrated over time, and appropriately rescaled and centered, it becomes possible to define a functional limit theorem for it, which then converges to a randomly pinned Brownian motion. By applying these theoretical results, a new goodness‐of‐fit test is derived. We demonstrate that this test is very efficient when it is applied to data which come from a multimodal or mixture distribution, like the classic Old Faithful dataset. The Canadian Journal of Statistics 46: 656–672; 2018 © 2018 Société statistique du Canada
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".