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The last result indicates that the error in the estimate of the integral decreases as the square root of the number of trials. This is a slow rate of convergence, and much of the modern work on Monte Carlo methods has been directed at either reducing the error in the sampling process or decreasing the number of trials necessary to achieve a desired accuracy.
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In most cases, the analyst is less interested in estimating the error than in determining how many trials must be made to achieve a desired accuracy. Fishman (1995) shows that, starting from Chebychev s inequality pr 1 |Z| 2 Z (17.10)
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de ning Z = p p and = / , and recalling that Var[Z] = Var[p] = p (1 p)/N , we obtain (17.11) pr[|p p| < ] 1 p (1 p)/(N 2 ) We must specify a con dence level, which is to say that the probability in Equation (17.11) must be greater than some desired level, say 90% or 95%. This is usually stated in the form (1 ) with = 0.1 or 0.05, respectively. It follows that the number of trials needed to achieve a desired con dence level is nC ( , , p) = p (1 p)/( 2 ) (17.12)
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where any real result is rounded up to the next integer. Equation (17.12) cannot be used as written because the value of p is not known a priori. One alternative is to use p as an estimate of p. An upper limit on nC follows from the recognition that p(1 p) has its maximum when p = 0.5. This gives nC ( , ) = 1/(4 2 ) (17.13)
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The subscript C indicates that it is based on the Chebychev approximation. An improved estimate follows from the Central Limit Theorem, which states that as N the quantity (NH Np)/[Np(1 p)]1/2 converges to the standard Normal distribution with mean of zero and standard deviation of 1. Fishman (1995) shows that the corresponding requirements for achieving a desired accuracy with a given con dence level are nN ( , , p) = p (1 p) [ nN ( , ) = [
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The expression 1 (x) is the inverse of the CDF for the standard Normal distribution; that is, it is the value of the variate that gives the cumulative distribution x. Any real results in Equations (17.14) and (17.15) are rounded up to the next integer. The subscript N indicates that the result is based on a Normal approximation. Fishman (1995) observes that this estimate does not converge uniformly so that many workers prefer to multiply its estimates by 2. Table 17.2 lists the maximum values of nC and nN for various error bounds and for 90% and 95% con dence levels. The values for nN are also plotted in Figure 17.2. The results are consistent with the results for the Hit-or-Miss strategy in Table 17.1. For 100 trials Table 17.2 and Figure 17.2 indicate that we can have 90% con dence that the error is less than 0.08 and 95% con dence that it is less than 0.1. Table 17.1 shows that in one out of the ten sets of 100 trials (Number 9) the mean differed from the true value by
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MONTE CARLO SIMULATION METHODS
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Monte Carlo Quadrature 100000 Number of points required 10000 1000 100 10 1 0.001 90% conf. 95% conf.
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Error in Monte Carlo estimate of integral
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Figure 17.2 Error estimates for Hit-or-Miss Monte Carlo integration. Table 17.2 accuracy Number of trials required to achieve desired
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90% con dence 0.005 0.01 0.02 0.03 0.04 0.05 0.1 0.2 0.3 0.4 0.5 nC 100,000 25,000 6,250 2,778 1,563 1000 250 63 28 16 10 nN 27,056 6,764 1,691 752 423 271 68 17 8 5 3
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95% con dence nC 200,000 50,000 12,500 5,556 3,125 2000 500 125 56 32 20 nN 38,415 9,604 2,401 1,068 600 385 97 25 11 4 4
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more than 0.1. For 1000 trials one would expect 90% con dence that the error is less than 0.025 and 95% con dence that it is less than 0.03. In fact, the error in the Hit-or-Miss strategy is 0.021 and in the Sample Mean strategy is about 0.004. It should be noted that Trial Number 9 in Table 17.1 presents an unusual result. It has the largest error observed in several repeated runs of the numerical experiment. However, it does represent the sort of result that can be expected in Monte Carlo simulation and is consistent with the calculated error bounds.