Batch variance estimators for the median of simulation output

Batching is a well-known technique for estimating the variance of point estimators computed from simulation experiments. The batch statistic variance estimator is simply the (appropriately scaled) sample variance of the original point estimator computed on subsets of data. The sample mean is the most common statistic used to estimate the steady-state mean of a simulation. In this note we show that the sample median has some attractive properties beyond those it has in the independent data setting (e.g., guarding against heavy-tail distributions). Specifically, we show that the sample median loses very little in efficiency for positively correlated output, and that the batch variance estimator of the sample median has comparable bias to that of the sample mean in estimating its asymptotic variance. For the rare case of negatively correlated data, however, the sample median is very inefficient relative to the sample mean. For ‘heavy-tail’ distributions the sample median retains its superiority over the sample mean.