Example 4.4 (A GLRT for the Hypothesis Testing Problem Defined in Example 4.3)
Recall Example 4.3 in which we showed that there is no UMP test for the following problem:
![images](/api/v2/epubs/9781118986769/files/images/eq4-80.png)
where y, s, and are all L-vectors and θ ≠ 0. The null hypothesis is simple with Θ0 = {0}, and the alternative hypothesis is composite with
. In Example 4.3, we showed that although we can find UMP tests if
or
, for the case of
, there is no UMP test. Moreover, for
, we cannot apply the LMP test formulation of (4.71) because it is not enough to consider a small neighborhood of θ such that either θ > 0 or θ < 0. Hence, we may instead attempt to derive a GLRT for this problem.
Since the null hypothesis is simple in problem (4.80), we only need to find the ML estimate of θ for the likelihood of y under H1:
Thus, ...
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