Use of Hotelling's T^2: Outlier Diagnostics in Mixtures
Bibliographic record
Abstract
Given Gaussian observation vectors $[\seqcl{\BY}{n}]$ having a common mean and dispersion matrix, a pervading issue is to identify shifted observations of type $\{\BYi\!\to\!\BYi\!+\!\bdeli\}.$ Conventional usage enjoins Hotelling's $\Tisq$ diagnostics, derived and applied under the mutual independence of $[\seqcl{\BY}{n}]$. Independence often fails, yet the need to identify outliers nonetheless persists. Accordingly, the present study reexamines $\Tisq$ under dependencies to include equicorrelations and more general matrices. Such dependencies are found in the analysis of calibrated vector measurements and elsewhere. In addition, mixtures of these distributions having star--shaped contours arise on occasion in practice. Nonetheless, the $\Tisq$ diagnostics are shown to remain exact in level and power for all such mixtures. Moreover, further matrix distributions, not necessarily having finite moments, are seen to generalize $n$--dimensional spherical symmetry to include non--Gaussian matrices of order $(n\!\times\!k)$ supporting $\Tisq.$ For these the use of $\Tisq$ remains exact in level. These findings serve to expand considerably the range of applicability of $\Tisq$ in practice, to include matrix Cauchy and other heavy tailed distributions intrinsic to econometric and other studies. Case studies serve to illuminate the methodology.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.033 | 0.147 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".