Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 46 / 1

Let $Z=(Z_1,\dots ,Z_p{)}^T$. Independence of the standard normal distributions gives $\mathbb{E}Z=0$, $\mathrm{Cov}(Z)=I_p$, and joint densityChoose a matrix $C$ with $C{C}^T=\Sigma$, for example the lower-triangular factor from the Cholesky decomposition. It exists and is nonsingular because $\Sigma$ is a symmetric positive-definite matrix. Then the required simulation isIt has mean $\mu$ and covariance matrix $C{C}^T=\Sigma$, and every linear combination of its coordinates is Gaussian, so it has the required multivariate normal distribution.

For the multivariate normal density, apply the change of variables formula to $z=C^{-1}(x-\mu )$. Its absolute Jacobian determinant is $|\det C{|}^{-1}=|\Sigma {|}^{-1/2}$, andHenceThe nonsingularity assumption is needed for a density with respect to $p$-dimensional Lebesgue measure.

Take the coordinate-selection matrixThen $A^{T}X=X_1$, $A^T\mu ={\mu }_1$ and $A^T\Sigma A={\Sigma }_{11}$. The permitted result on a linear image of a multivariate normal vector therefore givesThis is the marginal distribution obtained by discarding the other coordinates. The conclusion holds also for a positive-semidefinite marginal covariance: in that case the Gaussian law may be degenerate and need not have a full-dimensional density. If the original covariance is positive definite, its principal block ${\Sigma }_{11}$ is positive definite as well.

Assume an independent Gaussian sample, positive-definite population covariance, and $n>p$. Define the sample mean and unbiased sample covariance matrix byThe standard test is based on Hotelling's T-squared statisticReject the null hypothesis for large values. Under the null its exact distribution isThe factor uses the unbiased, divisor-$n-1$ covariance convention. Normal-sample theory makes $\bar{X}$ independent of the residual covariance, whose whitened version has a Wishart distribution; this yields the stated F-distribution. The condition $n>p$ gives invertibility of $S$ almost surely and positive denominator degrees of freedom.

This is also the likelihood-ratio test. To see the connection, put $W=(n-1)S$ and $\delta =\bar{X}-{\mu }_0$. The maximized covariance estimates under the unrestricted and null models are $W/n$ and $W/n+\delta {\delta }^T$, respectively. The determinant identity for a rank-one update givesThus the likelihood ratio is $(1+T^2/(n-1){)}^{-n/2}$, decreasing in $T^2$. The term best is understood here as this standard likelihood-ratio, affine-invariant procedure; a two-sided multidimensional alternative does not specify a single direction to optimize power against.

For the transformed sample $Y_i=A{X}_i+b$, with $A$ the prescribed nonsingular coordinate transformation, we haveConsequentlyThis proves the affine invariance of Hotelling's statistic. The statistic, its p-value and its test decision are unchanged by a nonsingular affine transformation, provided the null mean is transformed with the data. Changes of units and rotations are included.