Comparing two multivariate normal random variables

In summary, the conversation discusses the use of multivariate normally i.i.d random variables x and y, with size n vectors and variances of 1, to form vectors containing their means (mx and my). It is noted that if mx = my, then W = (mx - my)'(mx - my) follows a centrally chi square distribution with n degrees of freedom, and if mx ≠ my, then W follows a noncentral chi square distribution. The speaker is interested in finding out which vector is larger and questions whether comparing the noncentral chi square distributions of Wx and Wy is a viable method. It is clarified that this depends on the specific goal, whether it is a hypothesis test or estimating mx or my.
  • #1
rjjs
1
0
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.
 
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  • #2
rjjs said:
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?
 

1. What is a multivariate normal random variable?

A multivariate normal random variable refers to a statistical distribution that involves multiple variables that are normally distributed. This distribution is often used to model data that has multiple variables, such as in scientific experiments and studies.

2. How do you compare two multivariate normal random variables?

To compare two multivariate normal random variables, you can use statistical tests such as the Hotelling's T-squared test or the Mahalanobis distance. These tests can help determine if the two variables come from the same underlying distribution.

3. What factors should be considered when comparing two multivariate normal random variables?

Some factors to consider when comparing two multivariate normal random variables include the sample size, the number of variables, and the assumptions of normality and homogeneity of variance. It is also important to assess the relevance and significance of any differences observed between the variables.

4. Can two multivariate normal random variables be compared if they have different means and variances?

Yes, two multivariate normal random variables can still be compared even if they have different means and variances. The Hotelling's T-squared test, for example, can handle unequal variances as long as the sample sizes are equal.

5. How can comparing two multivariate normal random variables be useful?

Comparing two multivariate normal random variables can be useful in various contexts, such as in scientific research and data analysis. It can help identify any differences or similarities between the variables and provide insights into the relationship between them. This information can then be used to make informed decisions or draw conclusions.

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