Question

Let \( X_1, X_2, X_3 \) be a random sample from a bivariate normal distribution with unknown mean vector \( \mu \) and unknown variance-covariance matrix \( \Sigma \), which is a positive definite matrix. The p-value corresponding to the likelihood ratio test for testing \[ H_0: \mu = 0 \quad \text{against} \quad H_1: \mu \neq 0 \] based on the realization \[ \left\{ \left( 1, 2 \right), \left( 4, -2 \right), \left( -5, 0 \right) \right\} \] of the random sample equals _________ (round off to 2 decimal places).

Hint:

For likelihood ratio tests, calculate the likelihood ratio statistic and compare it with the distribution under the null hypothesis to compute the p-value.

Answer 1

Correct Answer: 1

For the likelihood ratio test, we calculate the test statistic and compare it with the critical value or compute the p-value. The formula for the likelihood ratio test statistic is: \[ \lambda = \frac{L(\hat{\mu}_0)}{L(\hat{\mu})} \] Using the given sample values and calculating the statistic gives a p-value of \( 1.00 \). Thus, the p-value is \( 1.00 \).