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, the finite‐sample bias is positive. Vats and Flegal [27] recommend
and
, which induces a positive bias of
offsetting the original bias in the opposite direction. For
, this estimator corresponds to the flat‐top batch means estimator of Liu and Flegal [28]. Under polynomial ergodicity and additional conditions on the batch size
, the lugsail batch means estimators are strongly consistent [26].
5 Stopping Rules
Monte Carlo simulations are often terminated according to a fixed‐time regime. That is, before the start of the simulation, it is decided that some steps of the process will be generated. The fixed‐time termination rule may be formally written as
(5)
By construction, , and simulation terminates when the criterion is met. The representation in Equation (5) allows further adjustments to our termination rule with an ‐fixed‐time approach, where for some , the simulation terminates at
(6)
The termination time is deterministically dependent on . Specifically, . Glynn and Whitt [29] show that as . However, since the structure of the underlying distribution and the quantity of interest are unknown, there is often little intuition as to what and should be for any given problem.
Alternatively, one could terminate according to a random‐time regime such as when the volume of a confidence region (possibly relative to some quantity) is below a prespecified threshold. These confidence region volumes, , could be either at Equation (2) or at Equation (3). Glynn and Whitt [29] and Vats et al. [26] show that the resulting confidence regions created at termination have the correct coverage, asymptotically. Since the simulation ends at a random time, the estimate of the limiting Monte Carlo variance–covariance matrix, used in construction of , is required to be strongly consistent. Glynn and Whitt [29] further show that weak consistency is not sufficient. We discuss stopping rules of this type for IID and MCMC sampling in the following sections.
5.1 IID Monte Carlo
The absolute precision sequential stopping rule terminates simulation when the variability in the simulation is smaller than a prespecified tolerance, . Specifically, simulation is terminated at time where
Here, ensures a minimal simulation effort. By definition, as . Thus, as the tolerance decreases, the required simulation size increases. The stopping rule explained in the motivating example in the introduction is a one‐dimensional absolute precision sequential stopping rule. This rule works best in small dimensions when each component is on the same scale and an informed choice of can be made (as in the motivating example).
In situations where the components of are in different units, stopping simulation when the variability in the estimator is small compared to the size of the estimate is natural. For a choice of norm , a relative‐magnitude sequential stopping rule terminates simulation at
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