Refer to Exercise 8.8 and 8.9. Show that the rejection region is equivalent

Refer to Exercise 8.8 and 8.9. Show that the rejection region λ ≤ λ_α is equivalent to the rejection region y̅ ≥ y̅_α, where P(λ ≤ λ_α) and P(y̅ ≥ y̅_α) = α. Use the fact that e^-a2 → 0 as |a| → ∞.  

Data from Exercise 8.8

Suppose y₁, y₂, ..... y_n is a random sample from a normal distribution with unknown mean m and variance σ² = 1, i.e.,

Show that the likelihood L of the sample is

Data from Exercise 8.9:

Suppose we want to test H₀: μ = 0 against the alternative H_a: μ > 0. Since the estimator of μ̂ = y̅ is , the likelihood ratio test statistic is

Show that

Transcribed Image Text:

1 f(y) %3D V27