Let X 1 ,X 2 , ,X n be a random sample from the beta distribution with and 1, 2 Show that the likelihood ratio test statistic for testing H 0 1 versus H 1 2 is a function of the statistic W n i 1 log X i n i 1 log (1 X i ) ...

Just the ordinary fact that W ni1 log X ni1 log1 X note that the logarithm is ta ...

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

Question:

Let X₁,X₂, . . .,X_n be a random sample from the beta distribution with α = β = θ and Ω = {θ : θ = 1, 2}. Show that the likelihood ratio test statistic Λ for testing H₀ : θ = 1 versus H₁ : θ = 2 is a function of the statistic W = Σⁿ_i=1 log X_i + Σⁿ_i=1 log (1 − X_i).