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final09_soln

final09_soln - E4708 Statistical Inference for FE Professor...

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Unformatted text preview: E4708. Statistical Inference for FE. Professor S. Kou. Final Exam, Aug 26, 2009. Closed Book Exam . Total 50 pts. Note: For &true or false¡questions, simply answer &true¡or &false¡. No explanation is needed. 1. (7 pts) Basic Facts from Empirical Finance. a. (2 pts) Null hypothesis: The defendant is innocent. Type I error: The defendant is innocent but the jury gives a guilty verdict. b. (1 pt) False. c. (2 pts) Kolmogorov-Siminov test and Jarque-Bera test. d. (1 pt)True e. (1 pt) False 2. (9 pts) The likelihoood for X 1 , ..., X n is given by f ( X j & ) = n Y i =1 e & & & X i X i ! : The prior density is & ¡ & 1 e & &=¢ &( ¡ ) ¢ ¡ : Therefore, the posterior is given by f ( & j X ) _ & ¡ & 1 e & &=¢ &( ¡ ) ¢ ¡ & n Y i =1 e & & & X i X i ! _ & ¡ & 1 e & &=¢ & n Y i =1 e & & & X i = & ¡ & 1+ P n i =1 X i exp & ¡ ¡ 1 ¢ + n ¢ & £ : Thus, the posterior density f ( & j X ) must be a gamma with parameter ~ ¡ and ~ ¢ , where ~ ¡ = ¡ + n X i =1 X i ; ~ ¢ = ¡ 1 ¢ + n ¢ & 1 : And we can use the mean of the posterior as a Bayesian estimator, i.e. the estimator is ^ & = ~ ¡ ~ ¢ = ¡ + P n i =1 X i 1 ¢ + n = ¡¢ 1 ¢...
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final09_soln - E4708 Statistical Inference for FE Professor...

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