#StackBounty: #normal-distribution #multivariate-analysis #t-distribution #hotelling-t2 Distribution of multivariate "$Z$-score&qu…

Bounty: 50

Suppose $mathbf{X}_1, dots, mathbf{X}_n sim N_p(mathbf{mu}, Sigma)$ where $mu in mathbb{R}^p$ and $Sigma$ is a $p times p$ covariance matrix.

Suppose $hat{Sigma}$ is the sample covariance matrix, and $bar{mathbf{X}}$ is the sample mean, then we know that

$$n(mathbf{bar{X}} – mu)^T hat{Sigma}^{-1}(mathbf{bar{X}} – mu) sim T^2_{p,n-1},, $$
where $T^2_{p,n-1}$ is the Hotelling T-squared distribution with dimensionality parameter $p$ and degrees of freedom $n-1$. Discussion on this can be found here. There is also an alternative $F$-distribution representation of the Hotelling $T^2$.

Q. Is there a known distributional form of $Y = sqrt{n}hat{Sigma}^{-1/2}(bar{mathbf{X}} – mu) $?

When $p = 1$, we know that $Y sim t_{n-1}$ distribution. However, for $p > 1$, from the description of the multivariate $t$ distribution here, $Y$ not distributed like a multivariate $t$ distribution.

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