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smid10_soln

# smid10_soln - E4702 Statistical Inference for Financial...

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Unformatted text preview: E4702. Statistical Inference for Financial Engineering. Professor S. Kou. Midterm, August 12, 2010. 11am-1:30pm. Closed Book Exam . Total 40 pts. Note: For &true or false¡questions, simply answer &true¡or &false¡. No explanation is needed. 1. (5 pts) The returns of these two managers are: A: p 1 & 60% + (1 ¡ p 1 ) & 15% : B: p 2 & 30% + (1 ¡ p 2 ) & 5% . When the Simpson¢s paradox appears, we must have p 1 & 60% + (1 ¡ p 1 ) & 15% < p 2 & 30% + (1 ¡ p 2 ) & 5% ; which gives us p 2 > 1 : 8 p 1 + 0 : 4 : Also, p 1 and p 2 must satisfy ¢ p 1 ;p 2 ¢ 1 : 2. (10 pts) a. (5 pts) The likelihood is L ( & ) = n Y i =1 e & & & X i X i ! : Thus log( L ( & )) = ( ¡ n& ) + log( & ) n X i =1 X i ¡ n X i =1 log( X i !) : @ @& log( L ( & )) = ¡ n + 1 & n X i =1 X i : Thus, the MLE for ^ & is given by 0 = ¡ n + 1 ^ & n X i =1 X i ; i.e. ^ & = P n i =1 X i n = & X: By the invariance principle, the maximum likelihood estimator for ¡ = (1 + & ) e & & is given by ^ ¡ , where ^ ¡ = (1 + ^ & ) e & ^ & = (1 + & X ) e & & X : b. (5 pts) Since ¡ = (1 + & ) e & & = P ( X = 0) + P ( X = 1) ; an unbiased estimator for ¡ is given by ~ ¡ = 1 n n X i =1 ( I f X i = 0 g + I f X i = 1 g ) ; where I f A g = 1 if A is true and I f A g = 0 if A is false. Indeed, we haveis false....
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