Refined normal approximations for the central and noncentral chi-square distributions and some applications
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Bibliographic record
Abstract
In this paper, we prove a local limit theorem for the chi-square distribution with r>0 degrees of freedom and noncentrality parameter λ≥0. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of r−2, which is significantly smaller than the maximal error bounds of order r−1/2 recently found by Horgan and Murphy [On the convergence of the chi square and noncentral chi square distributions to the normal distribution. IEEE Commun Lett. 2013;17(12):2233–2236. DOI:10.1109/LCOMM.2013.111113.131879] and Seri [A tight bound on the distance between a noncentral chi square and a normal distribution. IEEE Commun Lett. 2015;19(11):1877–1880. DOI:10.1109/LCOMM.2015.2461681]. Our results allow us to drastically reduce the number of observations required to obtain negligible errors in the energy detection problem, from 250, as recommended in the seminal work of Urkowitz [Energy detection of unknown deterministic signals. Proc IEEE. 1967;55(4):523–531. DOI:10.1109/PROC.1967.5573], to only 8 here with our new approximations. We also obtain an upper bound on several probability metrics between the central and noncentral chi-square distributions and the standard normal distribution, and we obtain an approximation for the median that improves the lower bound previously obtained by Robert [On some accurate bounds for the quantiles of a noncentral chi squared distribution. Stat Probab Lett. 1990;10(2):101–106. Available from: https://www.ams.org/mathscinet-getitem?mr=MR1072495].
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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 | 0.000 | 0.000 |
| 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 it