Approximate maximum likelihood estimation for one‐dimensional diffusions observed on a fine grid
Bibliographic record
Abstract
Abstract We consider a one‐dimensional stochastic differential equation that is observed on a fine grid of equally spaced time points. A novel approach for approximating the transition density of the stochastic differential equation is presented, which is based on an Itô‐Taylor expansion of the sample path, combined with an application of the so‐called ‐expansion. The resulting approximation is economical with respect to the number of terms needed to achieve a given level of accuracy in a high‐frequency sampling framework. This method of density approximation leads to a closed‐form approximate likelihood function from which an approximate maximum likelihood estimator may be calculated numerically. A detailed theoretical analysis of the proposed estimator is provided and it is shown that it compares favorably to the Gaussian likelihood‐based estimator and does an excellent job of approximating the exact, but usually intractable, maximum likelihood estimator. Numerical simulations indicate that the exact and our approximate maximum likelihood estimator tend to be close, and the latter performs very well relative to other approximate methods in the literature in terms of speed, accuracy, and ease of implementation.
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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 | 0.001 |
| 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 | 0.000 | 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".