Suppose (X₁,........, Xₙ) is a random sample from normal(0,σ²). The MLE in this case is σ = 1/n Σⁿ ᵢ=₁ X²ᵢ.
(a) Now derive the score test of H₀: σ² = σ²₀ versus H₁: σ² ≠ σ²₀. Express i tin terms of static T = nσ²/σ²₀.
(b) Now derive the generalized likelihood ratio test, again expressed in terms of T.
(c) Here, we know that nσ²/σ²₀ ∼ chi-square(n). Use this fact to derive an exact (equal-tails) test.[There actually is an UMPU test in this case, but it is not quite equal-tails because the distribution is not symmetric.]