Another look at bootstrapping the student t-statistic
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Let X, X 1, X 2, ... be a sequence of i.i.d. random variables with mean µ = EX. Let {v 1 () , ..., v () } =1 ∞ be vectors of nonnegative random variables (weights), independent of the data sequence {X 1, ..., X n } =1 ∞ , and put m n = Σ =1 v () . Consider v 1 () X 1, ..., v () X n , a bootstrap sample, resulting from re-sampling or stochastically re-weighing the random sample X 1, ..., X n , n ≥ 1. Put $$\bar X_n = \sum\nolimits_{i = 1}^n {X_i } /n$$ , the original sample mean, and define $$\bar X_{m_n }^* = \sum\nolimits_{i = 1}^n {v_i^{(n)} X_i /m_n }$$ , where m n := Σ =1 v () , the bootstrap sample mean. Thus, $$\bar X_{m_n }^* - \bar X_n = \sum\nolimits_{i = 1}^n {\left( {v_i^{(n)} /m_n - 1/n} \right)X_i }$$ . Put V 2 = Σ =1 (v () /m n − 1/n)2 and let S 2 , $$S_{m_n }^{*2}$$ respectively be the original sample variance and the bootstrap sample variance. The main aim of this exposition is to study the asymptotic behavior of the bootstrapped t-statistics $$T_{m_n }^* : = (\bar X_{m_n }^* - \bar X_n )/(S_n V_n )$$ and $$T_{m_n }^{**} : = \sqrt {m_n } (\bar X_{m_n }^* - \bar X_n )/S_{m_n }^*$$ in terms of conditioning on the weights via assuming that, as n → ∞, max1≤i≤n (v () /m n − 1/n)2/V 2 = o(1) almost surely or in probability on the probability space of the weights. In consequence of these maximum negligibility conditions on the weights, a characterization of the validity of this approach to the bootstrap is obtained as a direct consequence of the Lindeberg-Feller central limit theorem (CLT). This view of justifying the validity of bootstrapping i.i.d. observables is believed to be new. The need for it arises naturally in practice when exploring the nature of information contained in a random sample via re-sampling, for example. Conditioning on the data is also revisited for Efron’s bootstrap weights under conditions on n, m n as n → ∞ that differ from requiring m n /n to be in the interval [λ 1, λ 2] with 0 < λ 1 < λ 2 < ∞ as in Mason and Shao (2001). The validity of the bootstrapped t-intervals is established for both approaches to conditioning.
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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 | 0.013 | 0.020 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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