Relation between confidence regions and tests Let X F P W 2 be a statistical model Show the following (a) If C W X P is a confidence region with error level and 0 2 is chosen arbitrarily, then 0 62 C is the rejection region of a test of H0 W D 0 against H1 W 0 of level (b) Conversely, if for each 0 2 a non randomised level test of H0 W D 0 against H1 W 0 is given, then these tests can be combined to construct a confidence region with error level

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Question: Relation between confidence regions and tests. Let .X;F; P# W # 2 / be a statistical model. Show the following: (a) If C W X

Relation between confidence regions and tests. Let .X;F; P# W # 2 ‚/ be a statistical model. Show the following:

(a) If C W X ! P.‚/ is a confidence region with error level ˛ and #0 2 ‚ is chosen arbitrarily, then ¹#0 62 C./º is the rejection region of a test of H0 W # D #0 against H1 W # ¤ #0 of level ˛.

(b) Conversely, if for each #0 2 ‚ a non-randomised level-˛ test of H0 W # D #0 against H1 W # ¤ #0 is given, then these tests can be combined to construct a confidence region with error level ˛.

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