531f10MCIa

531f10MCIa - STAT 531: Monte-Carlo Integration HM Kim...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
STAT 531: Monte-Carlo Integration HM Kim Department of Mathematics and Statistics University of Calgary Fall 2010 1/19
Background image of page 1

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

View Full DocumentRight Arrow Icon
Numerical Solutions It is NOT always possible to analytically compute estimators associated with a given paradigm (maximum likelihood, Bayes, method of moments, etc). Estimators involve integrals with respect to probability distributions. The possibility of producing an almost infinity number of random variables distributed according to a given distribution give us access to the use of asymptotic results . Assessment of the convergence of simulation methods: probability results such as Law of Large Number or Central Limit Theorem . Fall 2010 2/19
Background image of page 2
Limiting Behaviour For a sequence of random variables X 1 , X 2 , ··· , the sequence converge in probability to the random variable X if lim n P ( | X n - X | < ε ) for every ε > 0. the sequence converges almost surely to X if P ( lim n | X n - X | < ε ) for every ε > 0. Fall 2010 3/19
Background image of page 3

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

View Full DocumentRight Arrow Icon
Suppose X n N ( 1 n , 1 n ) is a sequence of independent variables. Since the mean and variance are both converging to zero, intuitively one would expect that the sequence X n converges to X = 0 in some sense. In fact, it converges almost surely to zero. For iid sequences of one-dimensional random variables X 1 , X 2 , ··· , let X = X i n . the weak law of large number (WLLN) X converges in probability to μ = E ( X i ) if E ( | X i | ) < the strong law of large number (SLLN) X converges almost surely to μ = E ( X i ) if E ( | X i | ) < Fall 2010 4/19
Background image of page 4
If θ is a parameter and T n is a statistic based on X 1 , X 2 , ··· , X n , then T n is said to be weakly or strongly consistent for θ if T n converges in probability or almost surely to θ . T
Background image of page 5

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

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/04/2011 for the course STAT 531 taught by Professor Gaborlukacs during the Spring '11 term at Manitoba.

Page1 / 19

531f10MCIa - STAT 531: Monte-Carlo Integration HM Kim...

This preview shows document pages 1 - 6. Sign up to view the full document.

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