Rank statistics for jointly normal random vector

[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.

 

[mmwave]

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!