Wishart Distribution and Hotelling Tests - Multivariate Analysis - Statistics Previous Year Questions

Let $X_1, X_2$ and $X_3$ be independent and identically distributed $N_4(0, \Sigma)$ random vectors, where $\Sigma$ is a positive definite matrix. Further, let $\boldsymbol{X} = \begin{pmatrix} X_1^T \\ X_2^T \\ X_3^T \end{pmatrix}$ be a $3 \times 4$ matrix, where for any matrix $M, M^T$ denotes its transpose. If $W_m(n, \Sigma)$ denotes a Wishart distribution of order $m$ with $n$ degrees of freedom and variance-covariance matrix $\Sigma$, then which of the following statements is TRUE?

Let \( X_1, X_2, X_3, Y_1, Y_2, Y_3, Y_4 \) be independent random vectors such that \( X_i \) follows \( N_4(\underline{0}, \Sigma_1) \) distribution for \( i = 1, 2, 3 \), and \( Y_j \) follows \( N_4(\underline{0}, \Sigma_2) \) 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 \( \Sigma \) and \( I_4 \) denotes the \( n \times n \) identity matrix, then which one of the following statements is true?

Let \(X_1, X_2, \ldots, X_{20}\) be a random sample of size 20 from \(N_6(\mu, \Sigma)\), with \(\det(\Sigma) \neq 0\), and suppose both \(\mu\) and \(\Sigma\) are unknown. Let \[\bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad S = \frac{1}{19} \sum_{i=1}^{20} (X_i - \bar{X})(X_i - \bar{X})^T.\] Consider the following two statements: (I) The distribution of \(19 S\) is \(W_6(19, \Sigma)\) (Wishart distribution of order 6 with 19 degrees of freedom). (II) The distribution of \((X_3 - \mu)^T S^{-1} (X_3 - \mu)\) is \(\chi_6^2\) (Chi-square distribution with 6 degrees of freedom). Then which of the above statements is/are true?

Let \(\mathbf{X}_1, \mathbf{X}_2, \ldots, \mathbf{X}_{10}\) be a random sample of size 10 from a \(N_3(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) distribution, where \(\boldsymbol{\mu}\) and non-singular \(\boldsymbol{\Sigma}\) are unknown parameters. If
\[\bar{\mathbf{X}}_1 = \frac{1}{5} \sum_{i=1}^5 \mathbf{X}_i, \qquad \bar{\mathbf{X}}_2 = \frac{1}{5} \sum_{i=6}^{10} \mathbf{X}_i,\]
\[S_1 = \frac{1}{4} \sum_{i=1}^5 (\mathbf{X}_i - \bar{\mathbf{X}}_1)(\mathbf{X}_i - \bar{\mathbf{X}}_1)^T, \quad S_2 = \frac{1}{4} \sum_{i=6}^{10} (\mathbf{X}_i - \bar{\mathbf{X}}_2)(\mathbf{X}_i - \bar{\mathbf{X}}_2)^T,\]
then which one of the following statements is NOT true?

Let \(\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots, \boldsymbol{X}_{25}\) be a random sample of size \(25\) from a population having \(N_3(\boldsymbol{\mu}, \Sigma)\) distribution, where \(\boldsymbol{\mu}\) and non-singular \(\Sigma\) are unknown parameters. Let \(S = \frac{1}{24} \sum_{j=1}^{25} (\boldsymbol{X}_j - \bar{\boldsymbol{X}})(\boldsymbol{X}_j - \bar{\boldsymbol{X}})^T\), where \(\bar{\boldsymbol{X}} = \frac{1}{25} \sum_{j=1}^{25} \boldsymbol{X}_j\), and \(B = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 0 & 1 \end{bmatrix}\). Then which one of the following statements is true?