Page 539 - Probability and Statistical Inference

516 11. Likehood Ratio and Other Tests

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parameters are unknown and θ = (µ , µ , σ) ∈ ℜ×ℜ×ℜ . Given α ∈ (0, 1), we
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wish to find a level α LR test for choosing between a null hypothesis H : µ =
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µ and a two-sided alternative hypothesis H : µ ≠ µ . With n ≥ 2, let us denote
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for i = 1, 2. Here, is the pooled estimator of σ .
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Since H specifies that the two means are same, we have Θ = {(µ, µ, σ ):
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µ ∈ ℜ, σ ∈ ℜ }, and Θ = {(µ , µ , σ ) : µ ∈ ℜ, µ ∈ ℜ, σ ∈ ℜ }. The
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likelihood function is given by

Thus, we can write

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One should check that the maximum likelihood estimates of µ, σ obtained
from this restricted likelihood function turns out to be

Hence, from (11.3.3)-(11.3.4) we have

On the other hand, one has