5.3 Hotelling’s \(T^2\) distribution

Recall that in univariate statistics, Student’s \(t\)-distribution appears as the sampling distribution of \(\frac{\bar{x}-\mu}{s/\sqrt{n}}\), which is used for hypothesis tests and constructing confidence intervals.

Hotelling’s \(T^2\) distribution is the multivariate analogue of Student’s \(t\)-distribution. It plays an important role in multivariate hypothesis testing and confidence region construction, just as the Student \(t\)-distribution does in the univariate setting.

Definition 5.4 Suppose \(\bx \sim N_p(\bzero,\bI_p)\) and \(\bM \sim W_p(\bI_p,n)\) are independent, then the quantity \[\tau ^2 = n \bx^\top \bM^{-1} \bx\] is said to have Hotelling’s \(T^2\) distribution with parameters \(p\) and \(n\). We write this as \[\tau^2 \sim T^2(p,n).\]

This is reminiscent of the definition of the Student \(t\)-distribution: if \(x \sim N(0,1)\) and \(v\sim \chi^2_n\), then \[T = \frac{x}{\sqrt{v/n}} \sim t_n.\] Hotelling’s \(T^2\) distribution looks similar (albeit working with the square):a MVN random variable ‘divided’ by a Wishart r.v. divided by the degrees of freedom.

We can generalise the definition with the following result.

Proposition 5.11 Suppose \(\bx \sim N_p(\bmu,\bSigma)\) and \(\bM \sim W_p(\bSigma,n)\) are independent and \(\bSigma\) has full rank \(p\). Then \[ n (\bx-\bmu)^\top \bM^{-1} (\bx-\bmu) \sim T^2(p,n). \]

Proof. Define \(\by = \bSigma^{-1/2}(\bx-\bmu)\). Then, by Corollary 5.2, \(\by \sim N_p(\bzero,\bI_p)\). Further, let \(\bZ = \bSigma^{-1/2} \bM \bSigma^{-1/2}\) then \(\bZ \sim W_p(\bI_p,n)\) by applying 5.7 with \(\bA = \bSigma^{-1/2}\). From the definition, \(n \by^\top \bZ^{-1} \by \sim T^2(p,n)\) and \[\begin{eqnarray*} n \by^\top \bZ^{-1} \by &=& n (\bx-\bmu)^\top \bSigma^{-1/2} \bSigma^{1/2} \bM^{-1} \bSigma^{1/2} \bSigma^{-1/2} (\bx-\bmu) \\ &=& n(\bx-\bmu)^\top \bM^{-1}(\bx-\bmu) \end{eqnarray*}\] so the result is proved.

This result gives rise to an important corollary used in hypothesis testing when \(\bSigma\) is unknown.

Corollary 5.3 If \(\bar{\bx}\) and \(\bS\) are the mean and covariance matrix based on a sample of size \(n\) from \(N_p(\bmu,\bSigma)\) then \[ (n-1)(\bar{\bx}-\bmu)^\top \bS^{-1} (\bar{\bx}-\bmu) \sim T^2(p,n-1).\]

Proof. We have seen earlier that \(\bar{\bx} \sim N_p(\bmu,\frac{1}{n}\bSigma)\). Let \(\bx^\ast = n^{1/2} \bar{\bx}\) and let \(\bmu^\ast = n^{1/2} \bmu\). Then \(\bx^\ast=n^{1/2} \bar{\bx} \sim N_p(\bmu^\ast, \bSigma)\).

From Proposition 5.10 we know \(n\bS \sim W_p(\bSigma,n-1)\), and from Theorem 5.4 we know \(\bar{\bx}\) and \(\bS\) are independent. Applying Proposition 5.11 with \(\bx = \bx^\ast\) and \(\bM = n\bS\) we obtain \[ (n-1)(\bx^\ast - \bmu^\ast)^\top (n\bS)^{-1} (\bx^\ast - \bmu^\ast) \sim T^2(p,n-1),\] and given \(\bx^\ast - \bmu^\ast = n^{1/2} (\bx-\bmu)\) then \[\begin{align*} &(n-1)(\bx^\ast - \bmu^\ast)^\top (n\bS)^{-1} (\bx^\ast - \bmu^\ast)\\ & \qquad \qquad = (n-1)n^{1/2}(\bar{\bx}-\bmu)^\top n^{-1} \bS^{-1} n^{1/2}(\bar{\bx}-\bmu) \\ &\qquad \qquad = (n-1)(\bar{\bx}-\bmu)^\top \bS^{-1} (\bar{\bx}-\bmu). \end{align*}\]

Hotelling’s \(T^2\) distribution is not often included in statistical tables but the next result tells us that Hotelling’s \(T^2\) is a scale transformation of an \(F\) distribution.

Proposition 5.12 If \(\tau^2 \sim T^2(p,n-1)\) then \[\gamma^2 = \frac{n-p}{(n-1)p} \tau^2 \sim F_{p,n-p}.\]

Proof. Beyond the scope of the module.

We can apply this result to the previous corollary.

Corollary 5.4 If \(\tau^2 = (n-1)(\bar{\bx}-\bmu)^\top \bS^{-1} (\bar{\bx}-\bmu)\) then \[ \gamma^2 = \frac{n-p}{p} (\bar{\bx}-\bmu)^\top \bS^{-1} (\bar{\bx}-\bmu) \sim F_{p,n-p}. \]

We’ll use this result to do hypothesis tests.