smid10_soln

smid10_soln - E4702. Statistical Inference for Financial...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 falsequestions, simply answer &trueor &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 Simpsons 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....
View Full Document

This note was uploaded on 10/18/2010 for the course IEOR 4702 taught by Professor Kou during the Spring '10 term at Columbia.

Page1 / 5

smid10_soln - E4702. Statistical Inference for Financial...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online