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Combining Monte Carlo Estimators If I have many MC estimators, with/without various variance reduction techniques, which should I choose?

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Combining Estimators Suppose I have m unbiased estimators all of the same parameter Put these estimators in a vector Y Any linear combination of these estimators with coefficients that add to one is also an unbiased estimator of the parameter Which such linear combination is best?

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Best linear combination of estimators.

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Estimating Covariance

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Theorem on Optimal Linear Combination of estimators

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Example: combining the estimators of the call option price

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Example: (cont)

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MATLAB function OPTIMAL function [o,v,b,t1]=optimal(U) % generates optimal linear combination of five estimators and outputs % average estimator and variance. t1=cputime; Y1=(.53/2)*(fn(.47+.53*U)+fn(1-.53*U)); t1=[t1 cputime]; Y2=.37*.5*(fn(.47+.37*U)+fn(.84-.37*U))+.16*.5*(fn(.84+.16*U)+fn(1-.16*U)); t1=[t1 cputime]; Y3=.37*fn(.47+.37*U)+.16*fn(1-.16*U); t1=[t1 cputime]; intg=2*(.53)^3+.53^2/2; Y4=intg+fn(U)-GG(U); t1=[t1 cputime]; Y5=importance('fn','importancedens','Ginverse',U); t1=[t1 cputime]; X=[Y1' Y2' Y3' Y4' Y5']; mean(X) V=cov(X); Z=ones(5,1); C=inv(V); b=C*Z/(Z'*C*Z); o=mean(X*b); % this is mean of the optimal linear combinations t1=[t1 cputime]; v=1/(Z'*V1*Z); t1=diff(t1); % these are the cputimes of the various estimators.

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Results for option pricing [o,v,b]=optimal(rand(1,100000)) Estimators = 0.4619 0.4617 0.4618 0.4613 0.4619 o = 0.46151 % best linear combination (true value=0.46150) v = 1.1183e-005 %variance per uniform input b’ = -0.5503 1.4487 0.1000 0.0491 -0.0475

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Efficiency of Optimal Linear Combination Efficiency gain based on number of uniform random numbers 0.4467/0.00001118 or about 40,000. However, one uniform generates 5 estimators requiring 10 function evaluations. Efficiency based on function evaluations approx 4,000 A simulation using 500,000 uniform random numbers ; 13 seconds on Pentium IV (2.4 Ghz) equivalent to twenty billion simulations by crude Monte Carlo.

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Interpreting the coefficients b. Dropping estimators. Variance of the mean of 100,000 is Standard error is around.00001 Some weights are negative, (e.g. on Y1) some more than 1 (on Y2), some approximately 0 (could they be dropped? For example if we drop

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More examples Integrate the function (exp(u)-1)/(exp(1)-1), u is from 0 to 1 The efficiency gain is over 26000.

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