Question

Let \( X_1, X_2, X_3, Y_1, Y_2, Y_3, Y_4 \) be independent random vectors such that \( X_i \) follows \( N_4(0, \Sigma_1) \) distribution for \( i = 1, 2, 3 \), and \( Y_j \) follows \( N_4(0, \Sigma_2) \) distribution for \( j = 1, 2, 3, 4 \), where \( \Sigma_1 \) and \( \Sigma_2 \) are positive definite matrices. Further, let \[ Z = \Sigma_1^{-1/2} X X^T \Sigma_1^{-1/2} + \Sigma_2^{-1/2} Y Y^T \Sigma_2^{-1/2}, \] where \( X = [X_1 \, X_2 \, X_3] \) is a \( 4 \times 3 \) matrix, \( Y = [Y_1 \, Y_2 \, Y_3 \, Y_4] \) is a \( 4 \times 4 \) matrix and \( X^T \) and \( Y^T \) denote transposes of \( X \) and \( Y \), respectively. If \( W_m(n, \Sigma) \) denotes a Wishart distribution of order \( m \) with \( n \) degrees of freedom and variance-covariance matrix \( \Sigma \), then which one of the following statements is true?

• A. \( Z \) follows \( W_4(7, I_4) \) distribution
• B. \( Z \) follows \( W_4(4, I_4) \) distribution
• C. \( Z \) follows \( W_7(4, I_7) \) distribution
• D. \( Z \) follows \( W_7(7, I_7) \) distribution

Hint:

When dealing with the Wishart distribution, remember that the number of degrees of freedom and the covariance matrix determine the distribution type.

Answer 1

The Correct Option is A

We are given that \( X_1, X_2, X_3 \) follow a multivariate normal distribution \( N_4(0, \Sigma_1) \), and \( Y_1, Y_2, Y_3, Y_4 \) follow \( N_4(0, \Sigma_2) \). The matrix \( Z \) is a sum of Wishart-distributed terms.
Step 1: Wishart Distribution Properties.
The Wishart distribution \( W_m(n, \Sigma) \) is a generalization of the chi-squared distribution to multiple dimensions. For each term in the sum, we apply the properties of the Wishart distribution: \[ W_4(4, I_4) \quad \text{is the distribution of a matrix formed by a sum of independent Gaussian vectors.} \] Step 2: Determine the Distribution of \( Z \).
Given that \( Z \) involves the sum of two independent components (one related to \( X \) and the other to \( Y \)), the correct distribution for \( Z \) is \( W_4(4, I_4) \), where \( I_4 \) is the identity matrix.
Final Answer: \[ \boxed{W_4(4, I_4)}. \]