Uniformly Most Powerful Test - Example

Example

Let denote i.i.d. normally distributed -dimensional random vectors with mean and covariance matrix . We then have

which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being

Thus, we conclude that the test

\phi(T) = \begin{cases} 1 & \text{if } T > t_0 \\ 0 & \text{if } T < t_0 \end{cases}

is the UMP test of size for testing vs.

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