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✓ 3 results · all verified · 2 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Multivariate normal, Wishart, and Hotelling $T^{2}$ — Examples

1 · Prerequisites

2 · Summary

The examples show the deterministic rank obstruction when the sample size does not exceed the dimension, a numerical distinction between the exact Wishart trace and the covariance likelihood ratio, and a two-dimensional Hotelling calculation with its exact $F$ cutoff. The likelihood-ratio example explicitly avoids inferring a finite-sample null law from an observed statistic; the Hotelling example keeps the test decision conditional on the selected significance level.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

The unbiased sample covariance is singular when 2 ≤ n ≤ p

Example

For every dataset of $n$ vectors in $R^{p}$ with $2 \leq n \leq p$ , the covariance convention $S = R / (n - 1)$ from The iid multivariate-normal sample and covariance conventions is singular. Equality $n = p$ is included. At $n = 1$ , that covariance convention is undefined because its denominator is zero.

Facts & Assumptions

Given: Integers $2 \leq n \leq p$ and arbitrary vectors $x_{1}, \dots, x_{n} \in R^{p}$ .

[F1]

The centered scatter is $R = \sum_{i = 1}^{n} (x_{i} - \overset{ˉ}{x}) (x_{i} - \overset{ˉ}{x})^{T}$ and $S = R / (n - 1)$ . The iid multivariate-normal sample and covariance conventions

[F2]

Rank is the dimension of the image, nullity is the dimension of the kernel, and their sum equals the dimension of the finite-dimensional domain. Rank and nullity of a linear map with finite-dimensional domain Rank-nullity: $\dim_{F} V = nullity T + rank T$

Verification

1.1givenF1algebra

Let $v_{i} = x_{i} - \overset{ˉ}{x}$ . Their sum is $\sum_{i} v_{i} = 0$ , so the $n$ centered vectors are linearly dependent. Their span therefore has dimension at most $n - 1 \leq p - 1$ .

2.1F1F2step 1.1algebra

Let $V = [v_{1} ∣ \dots ∣ v_{n}] : R^{n} \to R^{p}$ . Since $V (1, \dots, 1)^{T} = 0$ , [F2] gives $rank V \leq n - 1 \leq p - 1$ , so $im V$ is a proper subspace of $R^{p}$ . The scatter matrix is $R = V V^{T}$ , hence $im R \subseteq im V$ ; it is not surjective and therefore is singular. Because $n - 1 > 0$ , $S = R / (n - 1)$ has the same non-surjective image and is singular.

3.1F1step 1.1step 2.1∎

If $n = p$ , the rank bound in step 1.1 is $p - 1$ , so equality of sample size and dimension still forces singularity. If $n = 1$ , the definition $S = R / (n - 1)$ has zero denominator and defines no sample covariance; this example makes no $n = 1$ covariance claim.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

The exact Wishart trace and covariance likelihood ratio are different statistics

Example

Take $p = 1$ , $n = 4$ , $Σ_{0} = 1$ , and the observed sample $(\sqrt{3 / 2}, - \sqrt{3 / 2}, 0, 0) .$ The exact covariance trace statistic and the covariance likelihood-ratio statistic have different values on this observation. The trace statistic has an exact $χ_{3}^{2}$ law under the null, whereas the likelihood-ratio result in The Gaussian covariance likelihood ratio has its fixed-dimensional Wilks limit gives only an asymptotic $χ_{1}^{2}$ calibration. This example does not assert a finite-sample law for the likelihood ratio.

Facts & Assumptions

Given: The displayed one-dimensional sample, $n = 4$ , and null variance $Σ_{0} = 1$ .

[F1]

For observations $x_{i}$ , $\overset{ˉ}{x} = n^{- 1} \sum_{i} x_{i}$ , $R = \sum_{i} (x_{i} - \overset{ˉ}{x})^{2}$ , $S = R / (n - 1)$ , and $M = R / n$ . The iid multivariate-normal sample and covariance conventions

[F2]

The null likelihood-ratio statistic is $n {tr (Σ_{0}^{- 1} M) - \log \det (Σ_{0}^{- 1 / 2} M Σ_{0}^{- 1 / 2}) - p}$ ; its chi-square result is asymptotic with $p$ fixed. The Gaussian covariance likelihood ratio has its fixed-dimensional Wilks limit

[F3]

Under the null, $tr (Σ_{0}^{- 1} (n - 1) S) \sim χ_{p (n - 1)}^{2}$ , including when the scatter is singular. The exact covariance trace statistic has p(n−1) chi-square degrees of freedom

Verification

1.1givenF1algebra

The sample has mean $0$ , so its centered sum of squares is $R = 3 / 2 + 3 / 2 = 3$ . Thus $S = R / 3 = 1$ and $M = R / 4 = 3 / 4$ by [F1].

2.1F2F3step 1.1algebra

Since $p = 1$ and $Σ_{0} = 1$ , the trace statistic is $(n - 1) S = 3$ , consistent with its $χ_{1 \cdot 3}^{2} = χ_{3}^{2}$ null law [F3]. Substituting $M = 3 / 4$ in [F2] gives the observed likelihood-ratio value $4 (\frac{3}{4} - \log \frac{3}{4} - 1) = 4 (\log \frac{4}{3} - \frac{1}{4}) \approx 0.1507.$ This is not the trace value $3$ .

3.1F2step 2.1∎

The calculations show two different finite-sample statistics and their distinct calibrations; the fact that one observed value is $0.1507$ makes no claim about the likelihood-ratio statistic's null distribution at $n = 4$ . The asymptotic qualification in [F2] remains necessary.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passOpen item page →

A two-dimensional Hotelling statistic and its exact F cutoff

Example

Assume AC and use the iid multivariate-normal sample model with $p = 2$ , $n = 4$ , and fixed positive-definite covariance $Σ$ . For the observation $x_{1} = (1, 0)^{T}$ , $x_{2} = (- 1, 0)^{T}$ , $x_{3} = (0, 1)^{T}$ , and $x_{4} = (0, - 1)^{T}$ , test $H_{0} : μ = μ_{0}$ with $μ_{0} = (1, 1)^{T}$ . The sample mean is zero, the sample covariance is $(2 / 3) I_{2}$ , the Hotelling statistic is $T^{2} = 12$ , and its scaled $F_{2, 2}$ statistic is $4$ . For a chosen $0 < α < 1$ , the exact test rejects precisely when $4 > F_{2, 2}^{- 1} (1 - α)$ .

Facts & Assumptions

Given: AC, the iid $N_{2} (μ, Σ)$ sample model with fixed $Σ ≻ 0$ , the four displayed observed vectors, $n = 4$ , $p = 2$ , $μ_{0} = (1, 1)^{T}$ , and $0 < α < 1$ .

[F1]

The sample covariance is $S = \sum_{i} (x_{i} - \overset{ˉ}{x}) (x_{i} - \overset{ˉ}{x})^{T} / (n - 1)$ . The iid multivariate-normal sample and covariance conventions

[F2]

Under AC, in the iid multivariate-normal model with $Σ ≻ 0$ and $n > p$ , $T^{2} = n (\overset{ˉ}{X} - μ_{0})^{T} S^{- 1} (\overset{ˉ}{X} - μ_{0})$ when $S ≻ 0$ and is zero otherwise. Under $μ = μ_{0}$ , $(n - p) T^{2} / (p (n - 1))$ has law $F_{p, n - p}$ . The one-sample Hotelling statistic has an exact F law

[F3]

Under the same model assumptions and $0 < α < 1$ , the test rejects when the scaled statistic exceeds the $1 - α$ quantile and has exact null size $α$ over all $Σ ≻ 0$ . The one-sample Hotelling F test has exact size alpha

[F4]

$F_{r, s}^{- 1} (u)$ is the unique $u$ -quantile of the F distribution for positive degrees of freedom. Student t and Fisher F laws Student t and Fisher F CDFs are continuous and strictly increasing on their supports

Verification

1.1givenF1algebra

The four vectors sum to zero, so $\overset{ˉ}{x} = 0$ . Their outer products sum to $2 I_{2}$ , hence $R = 2 I_{2}$ and $S = R / 3 = (2 / 3) I_{2}$ by [F1]. This matrix is positive definite and has inverse $(3 / 2) I_{2}$ .

2.1F2step 1.1algebra

The difference $\overset{ˉ}{x} - μ_{0} = (- 1, - 1)^{T}$ has squared Euclidean norm $2$ . Therefore [F2] gives $T^{2} = 4 (3 / 2) \cdot 2 = 12$ , and the scale $(n - p) / (p (n - 1)) = 2 / (2 \cdot 3) = 1 / 3$ makes the F statistic equal $4$ .

3.1F3F4step 2.1∎

By [F3]–[F4], applying the exact test rule to this observation is exactly the stated comparison $4 > F_{2, 2}^{- 1} (1 - α)$ . The example reports the rule as a function of $α$ and does not infer a rejection decision without choosing $α$ .

Multivariate normal, Wishart, and Hotelling T2T2 — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

The unbiased sample covariance is singular when 2 ≤ n ≤ p

Example

Facts & Assumptions

Verification

The exact Wishart trace and covariance likelihood ratio are different statistics

Example

Facts & Assumptions

Verification

A two-dimensional Hotelling statistic and its exact F cutoff

Example

Facts & Assumptions

Verification

Sources

Multivariate normal, Wishart, and Hotelling $T^{2}$ — Examples