Exercise 10 108 Suppose that X 1 , X 2 , , X n 1 , Y 1 , Y 2 , , Y n 2 and W 1 , W 2 , , W n 3 are independent random samples from normal distributions with respective unknown means 1 , 2 and 3 and variances 1 2 , 2 2 and 3 2 1 Find the likelihood ratio test for H 0 1 2 2 2 3 2 against the alternative of at least one inequality 2 Find an approximate critical region for the test in part (a) if n 1 , n 2 and n 3 are large and 0 05

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Question: Exercise 10.108: Suppose that X 1 , X 2 , ..., X n 1 , Y 1 , Y 2 , ..., Y n 2

Exercise 10.108:Suppose that X₁, X₂, ..., X_n₁, Y₁, Y₂, ..., Y_n₂ and W₁, W₂, ..., W_n₃ are independent random samples from normal distributions with respective unknown means ₁, ₂ and ₃ and variances ₁², ₂² and ₃².

1. Find the likelihood ratio test for H₀: ₁² = ₂² = ₃² against the alternative of at least one inequality.

2. Find an approximate critical region for the test in part (a) if n₁, n₂ and n₃ are large and = 0.05.

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