Testing that a multivariate mean approximately equals a vector of constants Suppose we have a $p$-variate random vector that has a multivariate Normal distribution, $\boldsymbol{X}\sim MVN(\boldsymbol{\mu},\Sigma)$. My hypothesis is that the mean vector contains only zeros, $\boldsymbol{\mu}=\boldsymbol{0}$. I think I can test that using Hotelling's $T^2$.
Now, I do not really believe my model is exactly correct, but I need it to be a good enough approximation. And I have a clear measure of what is good enough. I consider three cases of such a measure:

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*$\boldsymbol{\mu}^\top\boldsymbol{\mu}<c$,

*$\sum_{j=1}^p|\mu_j|<c$,

*$\max_{j\in 1,\dots,p}|\mu_j|<c$.

How would I test 1., 2. or 3.?

My first thought is to use a bootstrap approach.

*

*Take a bootstrap sample of your data.


*Calculate and store quantity $(1)$, $(2)$, or $(3)$ of your bootstrap sample


*Repeat, repeat, repeat...
Now you have a bunch of numbers. Use them to compute a confidence interval using one of the usual bootstrap methods (e.g., BCa).
If that entire confidence interval is lower than $c$, then you have statistical evidence that the population-level value of $(1)$, $(2)$, or $(3)$ is less than $c$. If you need a p-value, you might figure out the confidence level at which the upper limit of the confidence interval just touches $c$.
AS AN ASIDE, it might make more sense to take the square root of $(1)$. Then all three of your measures are $L_p$ norms and reasonably called distances between your mean vector and the zero vector. Then again, if you have reason to care about $(1)$ as it is written, go for it!
