href="#fb3_img_img_3aa5e4d9-0226-5d96-b997-c8fdff60270f.png" alt="script í’³"/>, and suppose that Thus, even when
The foundation of Monte Carlo simulation methods rests on asymptotic convergence as indicated by (1). When enough samples are obtained,
Although Monte Carlo simulation relies on large‐sample frequentist statistics, it is fundamentally different in two ways. First, data is generated by a computer, and so often there is little cost to obtaining further samples. Thus, the reliance on asymptotics is reasonable. Second, data is obtained sequentially, so determining when to terminate the simulation can be based on the samples already obtained. As this implies a random simulation time, additional safeguards are necessary to ensure asymptotic validity. This has led to the study of sequential stopping rules, which we present in Section 5.
Sequential stopping rules rely on estimating the limiting Monte Carlo variance–covariance matrix (when
Over a variety of examples in Section 7, we conclude that the simulation size required for a reliable estimation is often higher than what is commonly used by practitioners (see also Refs [6, 7]. Given modern computational power, the recommended strategies can easily be adopted in most estimation problems. We conclude the introduction with an example illustrating the need for careful sample size calculations.
Example 1. Consider IID draws
Confidence intervals are notoriously difficult to understand at a first instance, and thus a standard Monte Carlo experiment in an introductory statistics course is that of repeating the above experiment multiple times and illustrating that on average about
where
