Optimal Nested Simulation Experiment Design via Likelihood Ratio Method
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Nested simulation arises frequently in financial or input uncertainty quantification problems, where the performance measure is defined as a function of the simulation output mean conditional on the outer scenario. The standard nested simulation samples M outer scenarios and runs N inner replications at each. We propose a new experiment design framework for a problem whose inner replication's inputs are generated from probability distribution functions parameterized by the outer scenario. This structure lets us pool replications from an outer scenario to estimate another scenario's conditional mean via the likelihood ratio method. We formulate a bi-level optimization problem to decide not only which of M outer scenarios to simulate and how many times to replicate at each, but also how to pool these replications such that the total simulation effort is minimized while achieving the same estimation error as the standard nested simulation. The resulting optimal design requires far less simulation effort than MN. We provide asymptotic analyses on the convergence rates of the performance measure estimators computed from the experiment design. Empirical results show that our experiment design significantly reduces the simulation cost compared to the standard nested simulation as well as a state-of-the-art design that pools replications via regressions.
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