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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 10 Fall 2006 Issued: Thursday, November 9, 2006 Due: Thursday, November 16, 2006 Problem 10.1 (A non-parametric hypothesis test) A set of i.i.d. samples Y 1 ,...,Y n is drawn from an unknown distribution F . The null hypothesis H asserts that F has median μ = μ , while the alternative H 1 asserts that μ > μ . (Note that both hypotheses are composite, but neither is based on a parametric model, since only μ is given while the form of F is unknown.) (a) Consider the statistic S = ∑ n i =1 I [ Y i > μ ]. Compute the exact distribution of S under the null hypothesis. How could this be useful in performing a hypothesis test? (b) Consider the test δ s ( Y ) that rejects H when S exceeds some threshold s . Use asymp- totic theory to approximate the level α ( s ) = E [ δ s ( Y )] of this test under H as a function of s . (This test is known as a non-parametric sign test .) Problem 10.2Problem 10....
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- Fall '08
- Statistics, Null hypothesis, testing H0, UMP test