A bias-reduced approach to density estimation using Bernstein polynomials
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Bibliographic record
Abstract
Mixtures of Beta densities have led to different methods of density estimation for univariate data assumed to have compact support. One such method relies on Bernstein polynomials and leads to good approximation properties for the resulting estimator of the underlying density f. In particular, if f is twice continuously differentiable, this estimator can be shown to attain the optimal nonparametric convergence rate of n −4/5 in terms of mean integrated squared error (MISE). However, this rate cannot be improved upon directly when relying on the usual Bernstein polynomials, no matter what other assumptions are made on the smoothness of f. In this note, we show how a simple method of bias reduction can lead to a Bernstein-based estimator that does achieve a higher rate of convergence. Precisely, we exhibit a bias-corrected estimator that achieves the optimal nonparametric MISE rate of n −8/9 when the underlying density f is four times continuously differentiable on its support.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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