X x r so f 1 u log log1 u u 0 1 jimin ding math wustl

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x } , x R So F - 1 ( u ) = log( - log(1 - u )) , u (0 , 1) . Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 8
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Example 3: Generate a r.v. from F - 1 ( U ) Theorem 4.8.1 +Ex 4.8.1: If X F , then F ( X ) U [0 , 1] . And Y = F - 1 ( U ) F . For example, to generate X from f X ( x ) = exp { x - e x } , x R : (Extreme value distribution, log of Weibull distribution) It is easy to show the cdf is F X ( x ) = 1 - exp {- e x } , x R So F - 1 ( u ) = log( - log(1 - u )) , u (0 , 1) . Hence generate U U (0 , 1) , then X = log( - log(1 - U )) follows extreme value distribution with above pdf. Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 8
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Example 4: Monte Carlo CI Let X 1 , · · · , X n iid N ( μ, σ 2 ) . A 95% CI for μ : ¯ x ± t 0 . 025 ,n - 1 s/ n. This can be explained as if one construct 100 CIs, 95 of them will contain the true mean μ . Here 95% is called nominal coverage probability . Now we can construct a Monte Carlo simulation to check the actual coverage probability , and compare it with 95% nominal coverage probability. Algorithm: 1. Set k = 1 2. Generate X 1 , · · · , X n iid N ( μ, σ 2 ) 3. Construct a 95% CI: ¯ x ± t 0 . 025 ,n - 1 s/ n. 4. If k = N , go to step 5; otherwise, k = k + 1 , and go to step 2. 5. Count the number of CIs containing μ and denote as I . Then the actual coverage probability is 1 - ˆ α = I/N . Furthermore, the approximation error is 1 . 96 p ˆ α (1 - ˆ α ) /N .
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  • Fall '08
  • Staff
  • Monte Carlo method, Monte Carlo methods in finance, Markov chain Monte Carlo, Jimin Ding

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