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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:

  1. $\boldsymbol{\mu}^\top\boldsymbol{\mu}<c$,
  2. $\sum_{j=1}^p|\mu_j|<c$,
  3. $\max_{j\in 1,\dots,p}|\mu_j|<c$.

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

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  • $\begingroup$ Cool question! You would use a different $c$ in $(1)$ than $(2)$, right? // Particularly in $(1)$, you might consider something like (squared) Mahalanobis distance, rather than the usual notion of distance. Something similar could apply to $(2)$, but I do not know what. // I would be curious to know how robust proposed tests are to deviations for the multivariate normal assumption. $\endgroup$
    – Dave
    Commented Feb 21, 2023 at 18:45
  • $\begingroup$ @Dave, thanks! Yes, $c$ would be different in (1) vs. (2). I would only do one of the options, (1) or (2), not both, so I would not have to compare $c$ from one to $c$ from the other. Regarding distance, I am actually interested in (2), but I doubt it can be easily done, so I consider (1) as a simplification. I suppose it does not get simpler than that. Regarding the assumption of multivariate normality, that is a valid concern. But again, I am starting from the simplest possible case to probe what is possible. My actual problem is a little more complex than the one I have posted. $\endgroup$ Commented Feb 21, 2023 at 18:58
  • $\begingroup$ I wonder if some kind of bootstrap confidence ellipsoid would be useful here. Bootstrap your data, calculate either distance metric, and calculate a confidence ellipsoid. If that ellipsoid is contained within an acceptable distance (defined by $c$), then you're within the tolerance. (This makes sense in the univariate case, right?) $\endgroup$
    – Dave
    Commented Feb 21, 2023 at 19:09
  • $\begingroup$ @Dave, could you elaborate just a bit? I do not quite follow. $\endgroup$ Commented Feb 21, 2023 at 19:19
  • $\begingroup$ Maybe it's just a usual bootstrap. Resample your data, calculate $(1)$ or $(2)$, and repeat this over and over. Then you can use a bootstrap confidence interval of the many calculated metrics. If that confidence interval does not contain $c$, then you have statistical evidence that your vector is no more than $c$ away from the zero-vector. $\endgroup$
    – Dave
    Commented Feb 21, 2023 at 19:22

1 Answer 1

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My first thought is to use a bootstrap approach.

  1. Take a bootstrap sample of your data.

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

  3. 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!

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  • $\begingroup$ I look forward to other ideas, particularly if there are analytical solutions to this and how robust this bootstrap approach is to deviations from multivariate normality. $\endgroup$
    – Dave
    Commented Feb 21, 2023 at 19:49

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