# Multivariate confidence interval for means?

Let $\mathbf{X}=(X_1,X_2)$ a random sample, where $X_1=(X_{11},X_{21},\dots,X_{n1})$ and $X_2=(X_{12},\dots,X_{n2})$ are two random vectors. Find joint confidence intervals for $\mu_1$ and $\mu_2$ using Bonferroni and Hotelling method.

I'm not sure but, the Bonferroni method for the multivariate case will be

$$\overline{x}_j\pm \sqrt{\frac{p(n-1)}{n-p}F_{p,n-p}(\alpha)}\frac{s_j}{\sqrt{n}}$$

where $s_j$ is the sample standard error for the sample $j$ and $p$ is the number of random vectors, in this case $p=2$. Is right use the $F$ distribution or I need to use t-student?

What is the form of Hotelling confidence interval in multivariate case? Is it form elipse as confidence regions?

• Does that minus sign in front of $X_{21}$ really belong there? If so, what pattern of signs is intended? – whuber Oct 17 '16 at 14:29
• @whuber It was a mistake – user72621 Oct 17 '16 at 23:58