Multivariate Analysis - Statistics Previous Year Questions

Let \((X_1, X_2, X_3)\) be a trivariate normal random vector with mean vector \((-3, 1, 4)\) and variance-covariance matrix \[ \begin{pmatrix} 4 & 0 & 0 \\ 0 & 3 & -3 \\ 0 & -3 & 4 \end{pmatrix} \]. Which of the following statements are true? (i) \(X_2\) and \(X_3\) are independent. (ii) \(X_1 + X_3\) and \(X_2\) are independent. (iii) \((X_2, X_3)\) and \(X_1\) are independent. (iv) \(\frac{1}{2}(X_2 + X_3)\) and \(X_1\) are independent.

Let \(\underline{X}_1, \ldots, \underline{X}_n\) be a random sample of size \(n \ (\geq 2)\) from \(N_p(\underline{0}, \Sigma)\) distribution, where \(1 \leq p \leq n - 1\) and \(\Sigma\) is a positive definite matrix. Define \(\bar{\underline{X}} = \frac{1}{n} \sum_{i=1}^n \underline{X}_i \ \text{ and } \ (n - 1)S = \sum_{i=1}^n (\underline{X}_i - \bar{\underline{X}})(\underline{X}_i - \bar{\underline{X}})^T,\) where for any column vector \(\underline{U}, \underline{U}^T\) denotes its transpose. Then the distribution of the statistic \(T^2 = n \bar{\underline{X}}^T S^{-1} \bar{\underline{X}}\) is

Let the joint distribution of \( (X, Y) \) be bivariate normal with mean vector \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \) and variance-covariance matrix \( \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \), where \( -1 < \rho < 1 \). Let \( \Phi_{\rho}(0, 0) = P(X \leq 0, Y \leq 0) \). Then the Kendall's \( \tau \) coefficient between \( X \) and \( Y \) equals

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?

Suppose that \(\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots, \boldsymbol{X}_n, \boldsymbol{Y}_1, \boldsymbol{Y}_2, \ldots, \boldsymbol{Y}_n\) are independent and identically distributed random vectors each having \(N_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) distribution, where \(\boldsymbol{\Sigma}\) is non-singular, \(p > 1\) and \(n > 1\). If \(\bar{\boldsymbol{X}} = \frac{1}{n} \sum_{i=1}^n \boldsymbol{X}_i\) and \(\bar{\boldsymbol{Y}} = \frac{1}{n} \sum_{i=1}^n \boldsymbol{Y}_i\), then which one of the following statements is 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?