Shape Constrained Kernel PDF and PMF Estimation
Bibliographic record
Abstract
We consider shape constrained kernel-based probability density function (PDF) and probability mass function (PMF) estimation. Our approach is of widespread potential applicability and includes, separately or simultaneously, constraints on the PDF (PMF) function itself, its integral (sum), and derivatives (finite-differences) of any order. We also allow for pointwise upper and lower bounds (i.e., inequality constraints) on the PDF and PMF in addition to more popular equality constraints, and the approach handles a range of transformations of the PDF and PMF including, for example, logarithmic transformations (which allows for the imposition of log-concave or log-convex constraints that are popular with practitioners). Theoretical underpinnings for the procedures are provided. A simulation-based comparison of our proposed approach with those obtained using Grenander-type methods is favourable to our approach when the DGP is itself smooth. As far as we know, ours is also the only smooth framework that handles PDFs and PMFs in the presence of inequality bounds, equality constraints, and other popular constraints such as those mentioned above. An implementation in R exists that incorporates constraints such as monotonicity (both increasing and decreasing), convexity and concavity, and log-convexity and log-concavity, among others, while respecting finite-support boundaries via explicit use of boundary kernel functions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".