homework11 - Homework 11 Due Friday Nov 30 (1) Consider...

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Unformatted text preview: Homework 11 Due Friday Nov 30 (1) Consider problem 3 from Homework 10. Suppose the data are: 3.23 -2.50 1.88 -0.68 4.43 1.03 -0.07 -0.01 0.76 1.76 0.33 -0.31 0.30 -0.61 1.52 1.54 2.28 0.42 2.33 -1.03 0.39 0.17 3.18 5.43 4.00 Find the mle . Find the standard error using the delta method. Find the standard error using the parametric bootstrap. (2) Let Let and let be the mle. Create a data set (using ) consisting of n=100 observations. (2a) Use the delta-method to get the se and 95 percent confidence interval for . Use the parametric bootstrap to get the se and 95 percent confidence interval for . Use the nonparametric bootstrap to get the se and 95 percent confidence interval for Compare your answers. (2b) Plot a histogram of the bootstrap replications for the parametric and nonparametric bootstraps. These are estimates of the distribution of . The delta method also gives an approximation to this distribution namely, Normal( ). Compare these to the true sampling distribution of . Which approximation, parametric bootstrap, bootstrap, or delta method is closer to the true distribution? The mle is . (3) Let Generate a data set of size 50 with (3a) Find the distribution of . Compare the true distribution of to the histograms from the parametric and nonparametric bootstraps. (3b) This is a case where the nonparametric bootstrap does very poorly. In fact, we can prove that this is the case. Show that, for the parametric bootstrap but for the nonparametric bootstrap . Hint: show that, then take the limit as n gets large. (4) Suppose that 50 people are given a placebo and 50 are given a new treatment. 30 placebo patients show improvement while 40 treated patients show im1 3 e c a! fEG1db" 4 `X U 3 YW V 4 F 6 3 DE xw r 4 $ ygvuts0 3 " i 3 qh 3 A 75@ 3 6 4 3 86 4 9753 00 )$ )$ p1`U`) 3 4 g) 53 $ Q P H 20 3 R%T9SRIEG B C4 & 0) #! 21('&%$" 0 3 4 3 4 i $3 "h 4 T 4 0 3 "p i 3 "h $ 3 3 be the log-odds ratio. Note that if . Find the mle of . Use the delta method to find a 90 per cent confidence interval for . (4e) Use simulation to find the posterior mean and posterior 90 per cent interval for . (5) Let . Show that and (6) Consider the Bernoulli(p) observations 0101000000 Plot the posterior for using these priors: Beta(1/2,1/2), Beta(1,1), Beta(10,10), Beta(100,100). (7) Let F F F 0 E 2 F F 2 F $ F F F 0 $ ` 2 F $ 4 0 E 0 2 F 2$ T 0 2 $ T 4 gw 4 . Assume that where is known. Use the prior is the standard error of the mle . . 2 provement. Let where is the probability of improving under treatment and is the probability of improving under placebo. (4a) Find the mle of . Find the standard error and 90 per cent confidence interval using the delta method. Note: Let and . Treat as a single observation with probability function where is a Binomial. Compute the Fisher information from there is no need to divide the resulting standard error by . (4b) Find the standard error and 90 per cent confidence interval using the bootstrap. (4c) Use the prior . Use simulation to find the posterior mean and posterior 90 per cent interval for . (4d) Let 2F 9 ) F $4 T 4 # F F ) !" ) # 4 ' 4 ' 6 D F6 D 4 F I P D H 9 GDFCB C7 4 ) ) P! ED1CB A0 7 `$C@ 853 0 ) 9 7 4 4 F 0 F 3 $ F 6% 53 & 3 0 ( 3 $ & 22G ' & % 0 32 10$ )% F F ) 4 0 E $ 0 FU$ gATSg PS Q R % 6Q99EG F 4 0 $ ; (7a) Let be the prior. Show that the posterior is also a Gamma. Find the posterior mean. (7b) Find the Jeffreys' prior. Find the posterior. 0 $! X R"s! % U 3 ...
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This note was uploaded on 05/25/2008 for the course STAT 36625 taught by Professor Larrywasserman during the Fall '01 term at Carnegie Mellon.

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