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Uniformly Most Powerful Tests Definition: Let f (x|θ), θ ∈ Θ ⊂ Rp , be the joint pdf/pmf of X = (X1 , . . . , Xn ) ˜ ˜ (the parameter θ can be vector-valued) and let Θ0 be a nonempty proper subset of Θ. Then, a test rule ϕ(x) for testing H0 : θ ∈ Θ0 vs H1 : θ 6∈ Θ0 is called a ˜ uniformly most powerful (UMP) test of size α if 1. maxθ∈Θ0 Eθ ϕ(X ) = α ˜ 2. it holds that Eθ ϕ(X ) ≥ Eθ ϕ̃(X ) for all θ 6∈ Θ0 , given any other test rule ϕ̃(x) ˜ ˜ ˜ with maxθ∈Θ0 Eθ ϕ̃(X ) ≤ α. ˜ Methods of finding UMP Tests 1. Method I: Based on Neyman-Pearson Lemma 2. Method II: Using the Monotone Likelihood Ratio (MLR) property Method I (Neyman-Pearson Lemma-based) To find a UMP size α test for H0 : θ ∈ Θ0 vs H1 : θ 6∈ Θ0 , 1. first fix θ0 ∈ Θ0 (suitably) and also θ1 6∈ Θ0 2. then use the Neyman-Pearson lemma to find a MP size α test ϕ(x) for H0 : ˜ θ = θ0 vs H1 : θ = θ1 , where (a) ϕ(x) does not depend on θ1 6∈ Θ0 and ˜ (b) maxθ∈Θ0 Eθ ϕ(X ) = α ˜ Then ϕ(x) is a UMP size α test for H0 : θ ∈ Θ0 vs H0 : θ 6∈ Θ0 . ˜ 1 Example: Let X1 , . . . , Xn be a random sample from N(µ, 1), µ ∈ R. Find a UMP test of size α for H0 : µ ≥ µ0 vs H1 : µ < µ0 (where µ0 is a fixed real number). 2 Method II: Based on Monotone Likelihood Ratio, a property defined below Definition: Let f (x|θ), θ ∈ Θ ⊂ R, be the joint pdf/pmf of X = (X1 , . . . , Xn ) ˜ ˜ (note: the parameter θ is real-valued). Then, {f (x|θ) : θ ∈ Θ} is said to have ˜ monotone likelihood ratio (MLR) in a real-valued statistic T = t(X ) if: for any ˜ θ1 < θ2 , there exists a nondecreasing function gθ1 ,θ2 (·) : R → [0, ∞] such that f (x|θ2 ) ³ ´ ˜ = gθ1 ,θ2 t(x) , f (x|θ1 ) ˜ ˜ Here a/b = +∞ if a > 0, b = 0 ∀x ∈ {y : f (y |θ1 ) + f (y |θ2 ) > 0} ˜ ˜ ˜ ˜ Example: Let f (x|θ) = c(θ)h(x) exp[q1 (θ)t1 (x)], θ ∈ Θ ⊂ R be the joint pdf/pmf ˜ ˜ ˜ of X1 , . . . , Xn (i.e., in the exponential family) and q1 (θ) is nondecreasing. Show that {f (x|θ) : θ ∈ Θ} has MLR in t1 (x). ˜ ˜ Theorem: (Method II) Let f (x|θ), θ ∈ Θ ⊂ R, be the joint pdf/pmf of ˜ X1 , . . . , Xn . Assume that {f (x|θ) : θ ∈ Θ} has MLR in T = t(x). Then, ˜ ˜ 1. a size α UMP test for H0 : θ ≤ θ0 vs H1 : θ > θ0 (θ0 ∈ Θ fixed) is given by 1 if t(x) > k ˜ where γ ∈ [0, 1] and −∞ ≤ k ≤ ∞ are γ if t(x) = k ϕ(x) = constants satisfying Eθ0 ϕ(X ) = α. ˜ ˜ ˜ if t(x) < k 0 ˜ 3 2. a size α UMP test for H0 : θ ≥ θ0 vs H1 : θ < θ0 (θ0 ∈ Θ fixed) is given by 1 if t(x) < k ˜ where γ ∈ [0, 1] and −∞ ≤ k ≤ ∞ are γ if t(x) = k ϕ(x) = constants satisfying Eθ0 ϕ(X ) = α. ˜ ˜ ˜ if t(x) > k 0 ˜ Caution: Method II of finding a UMP test applies only when Θ ⊂ R (i.e., parameter θ is real-valued) and the testing problem is of the form “H0 : θ ≤ θ0 vs H1 : θ > θ0 ” or “H0 : θ ≥ θ0 vs H1 : θ < θ0 ” Example: Let X1 , . . . , Xn be a random sample from uniform(0,θ), θ > 0. Find a UMP test of size α for H0 : θ ≤ θ0 vs H1 : θ > θ0 (where θ0 > 0 is fixed). 4