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problem65

# problem65 - Problem 65(a If X =(X1 Xn is a random sample...

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Problem 65 ( a ) If X =( X 1 ,...,X n ) is a random sample from the PAR (1 , θ ) then the like- lihood function is L ( θ )= n Q i =1 θ x θ +1 i = θ n μ n Q i =1 x i ( θ +1) , θ > 0 and the log likelihood function is l ( θ n log θ ( θ +1) t, θ > 0 where t = n P i =1 log x i . The score function is S ( θ n θ t = n θ t θ . S ( θ )=0 if θ = n t .S i n c e S ( θ ) > 0 if 0 < θ < n t and S ( θ ) < 0 if θ > n t therefore by the First Derivative Test l ( θ ) has a absolute maximum at θ = n t . Thus ˆ θ = n t is the M.L. estimate of θ and ˆ θ = n T = n n P i =1 log X i is the M.L. estimator of θ . The likelihood ratio statistic for testing H : θ = θ 0 is 2log R ( θ 0 ; X )=2 h l ³ ˆ θ ; X ´ l ( θ 0 ; X ) i =2 h n log ˆ θ ³ ˆ θ +1 ´ t n log θ 0 +( θ 0 t i " n log Ã ˆ θ θ 0 ! + ³ θ 0 ˆ θ ´ t # . 1

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( b ) For the data n =25 and n P i =1 log x i =40 , ˆ θ = n t = 25 40 =0 . 625 the observed value of the likelihood ratio statistic for testing is H : θ =1 is 2log R (1;
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problem65 - Problem 65(a If X =(X1 Xn is a random sample...

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