# post8 - UMP Tests simple null against composite alternative...

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UMP Tests simple null against composite alternative H 0 : θ = θ 0 vs H 1 : θ 6 = θ 0 H 0 : θ = θ 0 vs H 1 : θ>θ 0 H 0 : θ = θ 0 vs H 1 : θ<θ 0 Uniformly Most Powerful Critical Region The critical region C is a uniformly most pow- erful critical region of size α for testing the simple H 0 against a composite H 1 if the set C is a best critical region of size α for testing H 0 against each simple hypothesis in H 1 . 1 Uniformly Most Powerful Test A test deﬁned by a uniformly most powerful critical region C of size α is called a UMP test, with signiﬁcance level α , for testing the simple H 0 against a composite H 1 . ? UMP tests do not always exist. ? “Uniformly” is associated with θ . C is a best critical region of size α for testing H 0 : θ = θ 0 against all θ values given by H 1 . 2 Example : (cont’d from § 6.4) X 1 , ··· ,X n f ( x ; θ )= 1 θ e - x/θ I (0 , ) ( x ) Example 2 in § 6.4, H 0 : θ =2vs H 1 : θ =4w ith X 1 ,X 2 C = { x 1 + x 2 9 . 5 } α =

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## This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue University-West Lafayette.

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