This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 7 Mariana OlveraCravioto Columbia University [email protected] February 8th, 2012 IEOR 4404, Simulation Lecture 7 1/18 How to choose g ( x ) The majorizing function g ( x ) must satisfy two purposes: I We must be able to generate from density h ( x ) = g ( x ) /c easily. I Our choice of g ( x ) should make the probability of rejection small (choose g ( x ) as close to f ( x ) as possible) In our Beta(4,3) we could have chosen g ( x ) as follows: f(x) x x g(x) 3/5 1 f(x) x g(x) 3/5 1 2.074 h(x) IEOR 4404, Simulation Lecture 7 2/18 Acceptancerejection method for discrete r.v. Suppose X is a discrete r.v. that takes values { x 1 ,x 2 ,x 3 ,... } and has PMF p ( x ) . Choose the majorizing function g ( x ) such that g ( x i ) ≥ p ( x i ) for all i = 1 , 2 , 3 ,... and set c = ∑ ∞ i =1 g ( x i ) . Define h ( x ) = g ( x ) /c . Algorithm: 1. Generate Y having PMF h ( x ) 2. Generate U ∼ Uniform[0,1], independent of Y 3. If U ≤ p ( Y ) /g ( Y ) , return X = Y . Otherwise, go back to step 1 and try again. IEOR 4404, Simulation Lecture 7 3/18 Pros and cons of the acceptancerejection method Disadvantages: I Requires more than one Uniform[0,1] random variates (at least two) to generate X . I A bad choice of the majorizing function g ( x ) may lead to a big probability of rejection, which means more random numbers will be needed. I We cannot know in advance how many iterations of the algorithm we are going to need to generate X . Advantages: I The method works even for very complicated distributions provided we know their PDF, f ( x ) , or PMF, p ( x ) . I The method works particularly well when X has finite support, since we can choose g ( x ) to be a constant (maybe not the best choice, but certainly simple to simulate from). IEOR 4404, Simulation Lecture 7 4/18 Composition method I Suppose we want to generate a r.v. X whose CDF F can be written as F ( x ) = α 1 F 1 ( x ) + α 2 F 2 ( x ) + ··· = ∞ X i =1 α i F i ( x ) where...
View
Full
Document
This note was uploaded on 03/18/2012 for the course IEOR 4404 taught by Professor C during the Spring '10 term at Columbia.
 Spring '10
 C

Click to edit the document details