531f10MCIa

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

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STAT 531: Monte-Carlo Integration HM Kim Department of Mathematics and Statistics University of Calgary Fall 2010 1/19

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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
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

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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
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

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

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

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