Let X 1 ,X 2 , , X n be a random sample from a ( 3, ) distribution, where 0 (a) Show that the likelihood ratio test of H 0 0 versus H 1 0 is based upon the statistic W n i 1 X i Obtain the null distribution of 2W 0 (b) For 0 3 and n 5, find c 1 and c 2 so that the test that rejects H 0 when W c 1 o...

a Using the binomial theorem we have 3 0415 so that W n i1 X1 Xn ...

Let X 1 ,X 2 , . . . , X n be a random sample from

Question:

Let X₁,X₂, . . . , X_n be a random sample from a Γ(α = 3, β = θ) distribution, where 0 < θ < ∞.
(a) Show that the likelihood ratio test of H₀ : θ = θ₀ versus H₁ : θ ≠ θ₀ is based upon the statistic W = Σⁿ_i=1 X_i. Obtain the null distribution of 2W /θ₀.
(b) For θ₀ = 3 and n = 5, find c₁ and c₂ so that the test that rejects H₀ when W ≤ c₁ or W ≥ c₂ has significance level 0.05.