Let X1,...,Xn be independently distributed, each with probability p or q as N(, 2 0) or N(,

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Let X1,...,Xn be independently distributed, each with probability p or q as N(ξ, σ2 0) or N(ξ, σ2 1).

(i) If p is unknown, determine the UMP unbiased test of H : ξ = 0 against K : ξ > 0.

(ii) Determine the most powerful test of H against the alternative ξ1 when it is known that p = 1 2 , and show that a UMP unbiased test does not exist in this case.

(iii) Let αk (k = 0,...,n) be the conditional level of the unconditional most powerful test of part (ii) given that k of the X’s came from N(ξ, σ2 0) and n − k from N(ξ, σ2 1). Investigate the possible values α0, α1,...,αn.

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Testing Statistical Hypotheses

ISBN: 9781441931788

3rd Edition

Authors: Erich L. Lehmann, Joseph P. Romano

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