# Rank statistics for jointly normal random vector

• rhz
In summary: Expert SummarizerIn summary, the conversation discusses a problem involving a jointly normal random vector and its mean and covariance. The goal is to find the probability that the index associated with the maximum element of the vector is the same as the index associated with the maximum element of the mean. This can be calculated using the properties of multivariate normal distributions. It is also possible to find the probability mass function of the index associated with the maximum element of the vector. Resources such as textbooks and research papers may provide more detailed solutions.
rhz
Hi,

Just found this forum. It would be fabulous if someone could point me in the direction of a solution to this problem.

Consider X, a jointly normal random vector of mean m and positive definite covariance C. I am interested in knowing the probability that the index associated with the maximum element of the random vector equals the index associated with the maximum element of the mean of the vector. More generally, it would be nice to know the probability mass function of the index associated with the maximum element of X.

Does anyone know where I could find the solution to this problem?

Thanks!

rhz.

Dear rhz,

Thank you for bringing this problem to our attention. It is an interesting question and certainly has practical applications in various fields such as statistics, finance, and engineering.

To address your question, we need to first understand the concept of jointly normal random vectors. This means that all the variables in the vector are normally distributed and their values are correlated with each other. In this case, the mean and covariance of the vector are important parameters to consider.

To find the probability that the index associated with the maximum element of the random vector equals the index associated with the maximum element of the mean, we can use the properties of multivariate normal distributions. Specifically, we can use the fact that the maximum element of a jointly normal random vector follows a normal distribution, with mean equal to the maximum element of the mean vector, and variance equal to the maximum element of the covariance matrix.

Using this information, we can then calculate the probability that the index associated with the maximum element of the random vector is the same as the index associated with the maximum element of the mean vector. This probability can be calculated using standard normal distribution tables or statistical software.

To find the probability mass function of the index associated with the maximum element of X, we can use the same approach. We can calculate the probability of each index being the maximum element of the vector, and then sum these probabilities to get the overall probability mass function.

In terms of resources, I would recommend looking into textbooks or online resources on multivariate normal distributions and their properties. You may also find articles or research papers that have addressed similar problems and provide a more detailed solution.

I hope this helps in your search for a solution to the problem. Best of luck!

## What is a jointly normal random vector?

A jointly normal random vector is a set of random variables that follow a multivariate normal distribution, meaning that each variable is normally distributed and their joint distribution is also normal.

## What is the purpose of rank statistics in analyzing a jointly normal random vector?

Rank statistics are used to determine the relationship between the variables in a jointly normal random vector. By ranking the observations of each variable, we can calculate the correlation between them and identify any patterns or trends in the data.

## How do you calculate the rank correlation coefficient for a jointly normal random vector?

The rank correlation coefficient, also known as Spearman's rank correlation coefficient, is calculated by first ranking the observations for each variable. Then, the differences between the ranks for each variable are squared and summed. Finally, the correlation coefficient is calculated as 1 minus 6 times the sum divided by n squared minus 1, where n is the number of observations.

## Can rank statistics be used with non-normal data?

Yes, rank statistics can be used with non-normal data. In fact, they are often used as an alternative to traditional parametric statistics when the data does not meet the assumptions of normality. This is because rank statistics are based on the ranks of the data rather than the actual values, making them more robust and less affected by outliers or non-normal distributions.

## What are some limitations of using rank statistics for a jointly normal random vector?

One limitation of using rank statistics for a jointly normal random vector is that they may not fully capture the relationship between the variables. Additionally, rank statistics may be less precise in detecting small changes in the data compared to parametric statistics. Finally, rank statistics may not be suitable for all types of analyses and may not provide as much information as other statistical methods.

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