*Bounty: 50*

*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, $Y$ is not distributed like a multivariate $t$ distribution.