Advanced Simulation: Importance Sampling and Variance Reduction
Apply importance sampling and advanced variance reduction techniques for Exam MAS-II.
Importance Sampling
Importance sampling changes the sampling distribution to concentrate samples where the integrand contributes most. To estimate E_f[h(X)] = integral of h(x)*f(x)dx, sample from an alternative distribution g and compute (1/n)*sum of h(X_i)*f(X_i)/g(X_i), where X_i are drawn from g. The likelihood ratio w(x) = f(x)/g(x) reweights samples. Optimal importance sampling uses g proportional to |h(x)|*f(x), minimizing variance. For rare event estimation (e.g., ruin probability), importance sampling shifts the distribution toward the rare event region, dramatically reducing the number of samples needed.
Other Variance Reduction Methods
Stratified sampling divides the sample space into strata and samples each proportionally or optimally. It guarantees variance reduction over simple random sampling. Latin hypercube sampling extends stratification to multiple dimensions by ensuring each marginal is uniformly represented. Control variates use a correlated random variable with a known expectation: the adjusted estimator theta-hat_CV = theta-hat minus c*(Y-bar minus E[Y]) reduces variance when Y is correlated with the estimand. Antithetic variates pair U and 1-U. Conditional Monte Carlo computes E[h(X,Y)] = E[E[h(X,Y)|Y]], replacing one source of randomness with its conditional expectation. Exam MAS-II tests these techniques in actuarial simulation contexts.