Bahadur–Kiefer Type Representations for Smoothed Conditional Quantile Estimators
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Bibliographic record
Abstract
Bahadur and Kiefer derived almost sure (a.s.) representations for the (unconditional) sample quantile function in terms of the standard (unsmoothed) empirical distribution function. Their representations later became commonly known as the Bahadur–Kiefer (BK) representations. In this article, we establish BK type a.s. representations, and the resulting laws of iterated logarithm, for three distinct fully nonparametric smooth conditional quantile estimators—with optimal orders for the remainders—viz. for a smooth linear type, a Parzen-type smoothed (integrated) inverse and a smooth inverse type (kernel) conditional quantile estimator (c.q.e.) under some broad conditions on the underlying cdf’s and the kernels and bandwidth sequences employed. We also demonstrate that of these the linear type c.q.e. is, in fact, ‘second-order-equivalent’ to the Parzen-type smoothed (integrated) inverse c.q.e. Some remarks are included on the comparative merits of these smooth c.q.e.’s, and their BK representations relative to their smooth and unsmoothed counterparts studied earlier in literature and possible extensions of the present results. Our results are of the exact a.s. type and provide improvements over those achieved hitherto in literature. They are of considerable value for studying the asymptotics of quantile regression analytics. AMS Subject Classification: Primary 62G05, 62G07; secondary: 60F15, 62G20, 62G30
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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 | 0.001 | 0.050 |
| 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) | 0.004 | 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 it