I am trying to solve this problem but I am not sure how to proceed to get the formula for the test statistic and critical values.

Suppose $X_1,X_2,\ldots X_n$ are i.i.d. observations from a multivariate normal distribution $N(\mu,\Sigma)$ where $\Sigma$ is known. Further assume that R is a given matrix and r a given vector. Use the likelihood ratio procedure to produce a test statistic for $$H_0\colon R\mu = r \\ H_1\colon R\mu \neq r$$ Give explicit formula for the test statistic and the critical values.

Can someone give me a hint how to proceed?

  • 5
    $\begingroup$ Hint: $\mathbf{Y} = \mathbf{RX}$ where $\mathbf{X} = (X_1, X_2, \ldots, X_n)^T$ is a normal vector with multivariate normal distribution $N(\mathbf{R}\mu,\mathbf{R}\Sigma\mathbf{R}^T)$ and away we go! $\endgroup$ – Dilip Sarwate Jun 29 '12 at 19:03

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