Question

Suppose that \( X_1, X_2, ...., X_n, Y_1, Y_2, ...., Y_n \) are independent and identically distributed random vectors each having \( N_p(\mu, \Sigma) \) distribution, where \( \Sigma \) is non-singular, \( p>1 \) and \( n>1 \). If \( \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \) and \( \overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i \), then which one of the following statements is true?

• A. There exists \( c>0 \) such that \( c (\overline{X} - \mu)^T \Sigma^{-1} (\overline{X} - \mu) \) has \( \chi^2 \)-distribution with \( p \) degrees of freedom
• B. There exists \( c>0 \) such that \( c (\overline{X} - \mu)^T \Sigma^{-1} (\overline{X} - \overline{Y}) \) has \( \chi^2 \)-distribution with \( (p - 1) \) degrees of freedom
• C. There exists \( c>0 \) such that \( c \sum_{i=1}^n (X_i - \overline{X})^T \Sigma^{-1} (X_i - \overline{X}) \) has \( \chi^2 \)-distribution with \( p \) degrees of freedom
• D. There exists \( c>0 \) such that \( c \sum_{i=1}^n (X_i - Y_i - \overline{X} + \overline{Y})^T \Sigma^{-1} (X_i - Y_i - \overline{X} + \overline{Y}) \) has \( \chi^2 \)-distribution with \( p \) degrees of freedom

Hint:

- Quadratic forms involving multivariate normal distributions often lead to \( \chi^2 \)-distributions.
- The degrees of freedom in the chi-square distribution correspond to the number of variables involved.

Answer 1

The Correct Option is A

1) Understanding the problem:
This is a standard result in multivariate statistics. The sample mean of independent random variables from a multivariate normal distribution follows a \( \chi^2 \)-distribution when scaled by the inverse covariance matrix.
2) Analysis of the options:
(A) Correct: This is the standard form for a quadratic form in a multivariate normal distribution. The expression \( (\overline{X} - \mu)^T \Sigma^{-1} (\overline{X} - \mu) \) follows a \( \chi^2 \)-distribution with \( p \) degrees of freedom.
(B) Incorrect: The degrees of freedom would be incorrect in this context, as the correct degrees of freedom would be \( p \).
(C) Incorrect: This expression describes the sum of squared deviations of the sample, which has \( p \) degrees of freedom but doesn't correspond to the exact expression in the question.
(D) Incorrect: This option introduces unnecessary terms involving \( Y \), which are not relevant in the context of this particular formulation.
The correct answer is (A).