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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 $α$ .

Depends on

Used by

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Sources

Arak Mathai, Serge Provost, and Hans J. Haubold, Multivariate Statistical Analysis in the Real and Complex Domains, Chapter 6 §§6.3–6.4, PDF pp. 20–42 (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)