hwk2_soln

hwk2_soln - IEOR E4702 Statistical Inference for Financial...

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Unformatted text preview: IEOR E4702 Statistical Inference for Financial Engineering HWK Solution 2 1. The likelihood L is given by L = n Y i =1 f ( X i j & ) = n Y i =1 e & & ( & ) X i X i ! Therefore, log L = n X i =1 ( & & + X i log & & log( X i !)) = & n& + log & n X i =1 X i & n X i =1 log( X i !) : Now @ log L @& = & n + 1 & n X i =1 X i ; @ 2 log L @& 2 = & 1 & 2 n X i =1 X i : Thus, the MLE is given by ^ & = 1 n n X i =1 X i ; and the Fisher information matrix I ( ¡ ) is given by I ( & ) = & E & @ 2 log L @& 2 ¡ = 1 & 2 E ( n X i =1 X i ) = n& & 2 = n & : Thus the 95% asymptotic c.i. for & is given by ^ & ¡ 1 : 96 ¢ I ( ^ & ) £ & 1 = ^ & ¡ 1 : 96 ¤ n ^ & ¥ & 1 = ^ & ¡ 1 : 96 n ^ &: 2. The likelihood is given by L = f ( X ; ¡ ) = n Y i =1 1 ¢ £ ¤ X i & ¤ ¢ ¥ = 1 ¢ n (2 ¥ ) n= 2 exp f& 1 2 ¢ 2 n X i =1 ( X i & ¤ ) 2 g : Therefore, log L ( ¡ ) = & log( ¢ n (2 ¥ ) n= 2 ) & 1 2 ¢ 2 n X i =1 ( X i & ¤ ) 2 : 1 Taking derivatives with respect to & , ¡ 2 , ¡ and then setting them to zero, we have...
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