# #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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