Let X 1 ,X 2 , ,X n be a random sample from a distribution with pdf f(x ) exp x 2(1 ), x , where 0 Suppose 1, 2 Consider the hypotheses H 0 2 (a normal distribution) versus H 1 1 (a double exponential distribution) Show that the likelihood ratio test can be based on the statistic W n i 1 (X 2 i X i...

We shall use the following facts 1 Jensens inequality If Ex then x EX x See ...

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

Question:

Let X₁,X₂, . . .,X_n be a random sample from a distribution with pdf f(x; θ) = θ exp{−|x|θ} /2Γ(1/θ), −∞ < x < ∞, where θ > 0. Suppose Ω = {θ : θ = 1, 2}. Consider the hypotheses H₀ : θ = 2 (a normal distribution) versus H₁ : θ = 1 (a double exponential distribution). Show that the likelihood ratio test can be based on the statistic W = Σⁿ_i=1(X²_i− |X_i|).