The Law of the Iterated Logarithm for the Error Distribution Estimator in First-Order Autoregressive Models
Bibliographic record
Abstract
This paper investigates the asymptotic behavior of kernel-based estimators for the error distribution in a first-order autoregressive model with dependent errors. The model assumes that the error terms form an α-mixing sequence with an unknown cumulative distribution function (CDF) and finite second moment. Due to the unobservability of true errors, we construct kernel-smoothed estimators based on residuals obtained via least squares. Under mild assumptions on the kernel function, bandwidth selection, and mixing coefficients, we establish a logarithmic law of the iterated logarithm (LIL) for the supremum norm difference between the residual-based kernel estimator and the true distribution function. The limiting bound is shown to be 1/2, matching the classical LIL for independent samples. To support the theoretical results, simulation studies are conducted to compare the empirical and kernel distribution estimators under various sample sizes and error term distributions. The kernel estimators demonstrate smoother convergence behavior and improved finite-sample performance. These results contribute to the theoretical foundation for nonparametric inference in autoregressive models with dependent errors and highlight the advantages of kernel smoothing in distribution function estimation under dependence.
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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 | 0.003 |
| 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".