Estimating the mean of a non-linear function of conditional expectation

Consider the problem of estimating the expectation of a non linear function of a conditional expectation. This function is allowed to be non-differentiable and discontinuous at a finite set of points to capture practical settings. We develop a nested simulation strategy to estimate this via simulation and identify bias and optimized mean square error allocation. We show that this mean square error converges to zero at the rate Γ^-2/3, as Γ → ∞, where Γ denotes the available computational budget. We also consider combining nested simulation technique with kernel based estimation methods. We note that while the kernel based method have a better convergence rate when the underlying random process has dimensionality less than or equal to three, pure nested simulation may be preferred when this dimension is above four.