Let Y 1 Y 2 Y n be the order statistics of a random sample from a uniform distribution on (0, ), where 0 (a) Show that for testing H 0 0 against H 1 0 is (Yn 0 ) n , Yn 0 , and 0 if Y n 0 (b) When H 0 is true, show that 2 log has an exact 2(2) distribution, not 2(1)

Question: Let Y 1 < Y 2 < < Y n be the order statistics of a random sample from a uniform distribution

Let Y₁ < Y₂ < · · · < Y_n be the order statistics of a random sample from a uniform distribution on (0, θ), where θ > 0.
(a) Show that Λ for testing H₀ : θ = θ₀ against H₁ : θ ≠ θ₀ is Λ = (Yn/θ₀)ⁿ, Yn ≤ θ₀, and Λ =0 if Y_n > θ₀.
(b) When H₀ is true, show that −2 logΛ has an exact χ2(2) distribution, not χ2(1).

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a Since Y1 Y2 Yn has an exact 2 distribution with n 1 degrees of freedom b lnY10 Y20 ... View full answer

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