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Unformatted text preview: 36-325/725 Homework 10 due Thursday November 21 (1) Let . We want to test that the median is versus . (a) Find the Wald test. (b) Explain how to do a nonparametric Wald test using the plug-in estimate and the bootstrap. (c) Find the likelihood ratio test. (d) Here is an exact (small-sample) nonparametric test. Let Let . Let Suppose we reject when where Show that this test has type I error that is less than or equal to . This is called the sign test. , and . First, (e) Let us compare these tests. Let simulate the test under . Do the following steps: (i) Simulate . (ii) Compute each test statistic and make the reject/retain decision. (iii) Repeat steps (i) amd (ii) 1,000 times and see how often you reject. Compare this to the nominal size . (f) To check the power of the test, repeat part(e) but simulate from a . Summarize your findings. (g) Let's check how robust the tests are to the assumption of Normality. Repeat (e) but simulate from a Cauchy distribution. Summarize your findings. 1 $ G t k hTWfg T 3 & ' t 5V x S p T 3 & 3 fed 2t p WT51# RC V & Q T S & Q U1# RC t p Q $G C 3 D B 3 C u S & ( y x w H Q X 0 dbave 3 u tr i H ec Y $ 2sG qp fbhgfdbaH ` X PIH G Q E C PIH G F3 DB $ G WlTkvji & v( & 93 0 #A@876( & 3 0 & #5421)( $" %#! , that is, (b) Analyze Mendel's data (example 11.16) using a test and using a likelihood ratio test. Report the test statistic and p-value in each case. (3) Repeat chapter 11 exercise 8. Treat this as a problem of comparing two multinomials. Find the likelihood ratio test and then use it to analyze the data. " $ 1 2 1 1& 4 p 3Pci 1 t 3 0 )( '&" %S $ & $%!& 9$ i #7 ( "!%& 3 $ i 1)( 3 0 $& 0 & $ i H ec H p fh v2dPa!i (2) Let and consider testing (a) Show that the likelihood ratio test is to reject when where (4) Repeat chapter 11 exercise 9b using FDR. (5) Chapter 12, exercise 1. .) 2 (6) Chapter 12, exercise 2. (Part (e) should read, find a 95 per cent interval for ...
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- Fall '02