hwk4_soln - IEOR E4702 Statistical Inference for Financial...

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IEOR E4702 Statistical Inference for Financial Engineering HWK Solution 4 1. a. Denote X i ’s the samples, i =1 ,...,n ,and O i ’s frequencies O i , i =1 ,..., 12 .W i th the assumption of geometric distribution and statistical independence of samples, the likelihood is L ( p )= n Y i =1 p X i 1 (1 p )= p P n i =1 X i n (1 p ) n , and ln L ( p )=( n X i =1 X i n )ln p + n ln(1 p ) . Taking derivative with respect to p and letting it to zero gives P n i =1 X i n p n 1 p =0 . Therefore, the estimator for p is ˆ p = P n i =1 X i n P n i =1 X i = 363 130 363 =0 . 6419 . b. There are 12 cells in the test. Rigorously speaking, we cannot directly apply the chi- square test, as some of the cells have very low frequencies. However, for this hwk problem, we shall ignore this issue. Let O i be the observed cell counts and E i be the expected cell counts under H 0 ,wh e r e under the geometric distribution E i = n ˆ p i 1 (1
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This note was uploaded on 10/18/2010 for the course IEOR 4702 taught by Professor Kou during the Spring '10 term at Columbia.

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hwk4_soln - IEOR E4702 Statistical Inference for Financial...

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