Question

Let \( X_1, X_2, \dots, X_{10} \) be a random sample of size 10 from a \( N_3(\mu, \Sigma) \) distribution, where \( \mu \) and non-singular \( \Sigma \) are unknown parameters. If \[ \overline{X_1} = \frac{1}{5} \sum_{i=1}^{5} X_i, \quad \overline{X_2} = \frac{1}{5} \sum_{i=6}^{10} X_i, \] \[ S_1 = \frac{1}{4} \sum_{i=1}^{5} (X_i - \overline{X_1})(X_i - \overline{X_1})^T, \quad S_2 = \frac{1}{4} \sum_{i=6}^{10} (X_i - \overline{X_2})(X_i - \overline{X_2})^T, \] then which one of the following statements is NOT true?

• A. $\frac{5}{6} (\overline{X_1} - \mu)^T S_1^{-1} (\overline{X_1} - \mu)$ follows an $F$-distribution with 3 and 2 degrees of freedom.
• B. $\frac{6}{5} (\overline{X_1} - \mu)^T S_1^{-1} (\overline{X_1} - \mu)$ follows an $F$-distribution with 2 and 3 degrees of freedom.
• C. $4(S_1 + S_2)$ follows a Wishart distribution of order 3 with 8 degrees of freedom.
• D. $5(S_1 + S_2)$ follows a Wishart distribution of order 3 with 10 degrees of freedom.

Hint:

- The sum of independent Wishart-distributed random variables results in a new Wishart distribution with degrees of freedom equal to the sum of the degrees of freedom of the individual distributions.
- Scaling a Wishart-distributed random variable by a constant does not change its degrees of freedom.

Answer 1

The Correct Option is D

1) Understanding the distributions:
- $S_1$ and $S_2$ are sample covariance matrices computed from two independent subsets of the data. Since $S_1$ and $S_2$ are based on 5 observations each, the sum $S_1 + S_2$ is a sum of two Wishart distributions with 5 degrees of freedom each.
- The total degrees of freedom of $S_1 + S_2$ is thus $5 + 5 = 10$.
- Therefore, $S_1 + S_2$ follows a Wishart distribution with 10 degrees of freedom and order 3.
2) Analyzing the options:
- Option (A): $\frac{5}{6} (\overline{X_1} - \mu)^T S_1^{-1} (\overline{X_1} - \mu)$ follows an $F$-distribution with 3 and 2 degrees of freedom. This is correct, as this form follows the definition of an $F$-distribution.
- Option (B): $\frac{6}{5} (\overline{X_1} - \mu)^T S_1^{-1} (\overline{X_1} - \mu)$ follows an $F$-distribution with 2 and 3 degrees of freedom. This is also correct.
- Option (C): $4(S_1 + S_2)$ follows a Wishart distribution of order 3 with 8 degrees of freedom. This is correct because the sum of two Wishart distributions results in the total degrees of freedom of 10, but scaling the Wishart distribution by a constant does not affect the degrees of freedom.
- Option (D): $5(S_1 + S_2)$ follows a Wishart distribution of order 3 with 10 degrees of freedom. This is incorrect because $S_1 + S_2$ already has 10 degrees of freedom, and multiplying by 5 does not change the degrees of freedom, which remain 10.
Thus, the correct answer is (D).