Comparing two multivariate normal random variables

[rjjs]

I have two multivariate normally i.i.d random variables, x and y, that are size n vectors. Let us assume for simplicity that their variances are 1. From these random variables, I form two vectors that contain their means, and denote these mx and my.

I know that if mx = my, then W = (mx - my)'(mx - my) is centrally chi square distributed with n degrees of freedom. If mx ≠ my, then the distribution of W is noncentral chi square. Hence, by locating W at the central chi square distribution, I can place a confidence level for mx = my.

In addition to knowing whether mx = my, I want to find out which one, mx or my, is larger in some sense. For example, taking Wx = mx'mx and Wy = my'my would lead to comparing two noncentral chi squared distributions with df n and noncentrality parameters mx'mx and my'my.

The question is, is this a way to go with finding out whether mx > my or my > mx, and how to proceed from here? I'm confused about what relevant information the noncentral chi square distributions of Wx and Wy provide, and how to actually compare these? They are altogether different distributions because they have different noncentrality parameters.

 

[Stephen Tashi]

I have two multivariate normally i.i.d random variables, x and y, that are size n vectors.

I can place a confidence level for mx = my.

What is the definition of the confidence interval for which you seek the confidence level?
Are you doing a hypothesis test for "mx = my" or are you trying to create a confidence interval for estimating mx or my?

The question is, is this a way to go with finding out whether mx > my or my > mx, and how to proceed from here?

Since mx and my are vectors, you should define what the notation "mx > my" indicates. Does it indicated a component-by-component comparison?