The variance reduction methods in OSP-4~8 involve several choices and tuning parameters
- Control Variables: choosing the control variable
- Importance Sampling: choosing the proxy density (from intuition, maximum principle, WLS principle)
- Conditional Monte Carlo: choosing the conditioning random variable
- Stratified Sampling: choosing the strata and the number of strata
Suppose are Monte Carlo estimators obtained from different methods or the same method with multiple choice of parameters, then we can consider a combined estimator
such that (to make sure combined estimator unbiased).
The variance of this estimator is . The optimizing vector that yields minimum variance is
If the estimators are independent (or just uncorrelated) so that is a diagonal matrix, then optimal combination estimator is
The optimal choice depends on (the covariance matrix of ), which we do not have access to. We can use cross-fitting or leave-one-out like the control variables methods.
If , where are independent random vectors, then
We can use this covariance matrix estimator to combine the estimators. Besides, it is recommended that use the same set of uniform random numbers to implement all the methods when possible.
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