Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Chi-square laws, including zero degrees
Definition
Assume the Axiom of Choice. For an integer , let have law , so its coordinates are independent random variables. The chi-square law with degrees of freedom, denoted , is the law of
For , define , the point mass at zero. This zero-degree law records a rank-zero projection only; it is not used as a denominator in a Student or Fisher ratio. The multivariate-normal characteristic-function theorem makes the definition independent of the chosen realization of .
Depends on
Used by
- A boundary null has a nonstandard likelihood-ratio limit Counterexample
- Student t and Fisher F laws Definition
- A one-sided normal likelihood ratio at the boundary Example
- A skewed finite-replicate law defeats a nominal Student interval Example
- Known-variance normal mean and exact likelihood calibration Example
- Positive-degree chi-square laws have a positive density and regular CDF Lemma
- The standard Wishart inverse quadratic form has an exact Fisher law Lemma
- Exact confidence and prediction intervals in Gaussian regression Theorem
- Exact t and partial F inference in full-rank Gaussian regression Theorem
- Fixed-order autoregressive OLS martingale limit Theorem
- Normal sample mean and residual are independent, giving the exact t pivot Theorem
- Orthogonal projections split a standard Gaussian into independent chi-square sums Theorem
- QR decomposition gives exact t and F pivots in the fixed Gaussian linear model Theorem
- The exact covariance trace statistic has p(n−1) chi-square degrees of freedom Theorem
- The Gaussian covariance likelihood ratio has its fixed-dimensional Wilks limit Theorem
- The one-sample Hotelling statistic has an exact F law Theorem
- Wald, score, and likelihood-ratio asymptotic chi-square calibration Theorem
Dependency tree · two levels
27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Yen-Chi Chen, UW STAT 512 Lecture 8: Linear Models and Multivariate Normal Distributions (standard reference, not scraped)
- Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2003, Lectures 16–17 (standard reference, not scraped)