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Unformatted text preview: 98 Solutions Manual for Statistical Inference 9.25 The confidence interval derived by the method of Section 9.2.3 is C(y) = : y + 1 1 log y + log 1  n 2 n 2 where y = mini xi . The LRT method derives its interval from the test of H0 : = 0 versus H1 : = 0 . Since Y is sufficient for , we can use fY (y  ). We have (y) = sup=0 L(y) sup(,) L(y) = nen (y  0 )I[0 ,)(y) ne(yy) I[,)(y) 0 en(y0 ) if y < 0 if y 0 . = en(y0 ) I[0 ,) (y) = We reject H0 if (y) = en(y0 ) < c , where 0 c 1 is chosen to give the test level . To determine c , set = P { reject H0  = 0 } = P = P = Y > 0  0  log c n Y > 0  log c or Y < 0 = 0 n nen(y0 ) dy log c = 0 n =
0  log c n en(y0 ) = elog c = c . Therefore, c = and the 1  confidence interval is C(y) = : y  log n = : y + 1 log y . n To use the pivotal method, note that since is a location parameter, a natural pivotal quantity is Z = Y  . Then, fZ (z) = nenz I(0,) (z). Let P {a Z b} = 1  , where a and b satisfy = 2
a nenz dz = enz
0 a 0 = 1  ena = 2 nenz dz = enz
b b = enb 2  log 1  a= n nb = log 2 1 b =  log n 2 ena = 1  2 Thus, the pivotal interval is Y + log(/2)/n Y + log(1  /2), the same interval as from Example 9.2.13. To compare the intervals we compare their lengths. We have Length of LRT interval Length of Pivotal interval = y  (y + = 1 1 log ) =  log n n 1 1 y + log(1  /2)  (y + log /2) = n n 1 1  /2 log n /2 Thus, the LRT interval is shorter if  log < log[(1  /2)/(/2)], but this is always satisfied. 9.27 a. Y = Xi gamma(n, ), and the posterior distribution of is (y) = (y + 1 )n+a 1 1 1 b e (y+ b ) , (n + a) n+a+1 ...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
 Spring '12
 Dr.Hackney
 Statistics

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