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The exact covariance trace statistic has p(n−1) chi-square degrees of freedom

Statement

Assume AC, $p \geq 1$ , $n \geq 2$ , and the iid multivariate-normal sample model. Under $H_{0} : Σ = Σ_{0}$ for a specified $Σ_{0} ≻ 0$ , the distinct statistic $tr {Σ_{0}^{- 1} (n - 1) S}$ has the exact law $χ_{p (n - 1)}^{2}$ , even when $n \leq p$ and $S$ is singular. This is the Wishart trace statistic, not the covariance likelihood-ratio statistic.

Facts & Assumptions

Given: AC, $n \geq 2$ , $p \geq 1$ , the iid Gaussian sample model, and true covariance $Σ_{0} ≻ 0$ .

[A1]

AC is used in the Gaussian law and Wishart construction. The Axiom of Choice

[F1]

The centered scatter satisfies $(n - 1) S \sim W_{p} (Σ, n - 1)$ for every $n \geq 2$ . The Gaussian sample mean is independent of the Wishart sample scatter

[F2]

The Wishart law is the law of a sum of independent centered Gaussian outer products, including singular support. Wishart law by Gaussian outer products

[F3]

Under the null, each $N_{p} (0, Σ_{0})$ vector has the representation $Σ_{0}^{1 / 2} Z_{j}$ with independent standard-normal coordinates. The positive square root exists by A non-negative operator has a unique non-negative square root; if $Σ_{0}^{1 / 2} v = 0$ , then $v^{T} Σ_{0} v = ∥ Σ_{0}^{1 / 2} v ∥^{2} = 0$ , so $Σ_{0} ≻ 0$ makes it invertible. Multivariate normal law, including singular covariance

[F4]

The chi-square law with positive integer $k$ is the law of a sum of $k$ independent squared standard normals. Chi-square laws, including zero degrees Independent random elements have product joint law

Proof

1.1A1F1F2F3given

Put $m = n - 1 \geq 1$ . By [F1]–[F3], the law of $(n - 1) S$ is the law of $\sum_{j = 1}^{m} X_{j} X_{j}^{T}$ for independent $X_{j} = Σ_{0}^{1 / 2} Z_{j}$ with all $p m$ scalar coordinates independent standard normals.

2.1F3step 1.1algebra

For each summand, cyclicity of the finite trace and $Σ_{0}^{- 1 / 2} Σ_{0}^{1 / 2} = I_{p}$ give $tr (Σ_{0}^{- 1} X_{j} X_{j}^{T}) = X_{j}^{T} Σ_{0}^{- 1} X_{j} = Z_{j}^{T} Z_{j} .$ By linearity of trace, the target statistic therefore has the law of $\sum_{j = 1}^{m} \sum_{a = 1}^{p} Z_{j a}^{2}$ .

3.1F4step 2.1algebra∎

The $p m = p (n - 1)$ coordinates in step 2.1 are independent standard normals, so their squared sum has law $χ_{p m}^{2}$ by [F4]. This argument does not require $m \geq p$ : if $m < p$ , the matrix scatter may be singular, but its trace and the sum of squares remain defined; $m = 1$ and $p = 1$ are both included. The statistic is distinct from the likelihood-ratio statistic, whose formula contains a log determinant when the scatter is nonsingular.

Depends on

Used by

The exact Wishart trace and covariance likelihood ratio are different statistics Example

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Sources

Arak Mathai, Serge Provost, and Hans J. Haubold, Multivariate Statistical Analysis in the Real and Complex Domains, Chapter 5 §5.5, PDF pp. 47–66 (standard reference, not scraped)
Soumendu Mukherjee, Multivariate Analysis 2021, Lecture 3, PDF pp. 1–5 (standard reference, not scraped)
Peter D. Hoff, Multivariate Statistical Analysis, Duke STA 832 lecture notes, relevant sections in Chapters 4 and 7 (standard reference, not scraped)