Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes
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Bibliographic record
Abstract
Let X 1,..., X n, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H 0: X 1,..., X n is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H 0 versus the “at most one change (AMOC) in the mean” alternative hypothesis H A: there is an integer k*, 1 ≤ k* < n, such that EX 1 = · · · = EXk* ≠ EXk*+1 = ··· = EX n. A natural way of testing for H 0 versus H A is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δk = |S k/k - (S n - S k)/(n - k)|, where S k:= X 1 + ··· + X k. Asymptotic distributions of these test statistics under H 0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Z n(t)|, where {Z n(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that $${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n $$ if 0 ≤ t < 1, and Z n(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H 0 versus H A via sup-functionals of appropriately weighted |Z n(t)|. Simply considering max1ɤk<n Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Z n(t)|/(t(1 - t)) under H 0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Z n(t)| has not been considered before.
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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 | 0.006 | 0.102 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 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)
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