Overlapping batch quantiles

We show that although overlapping batch quantiles (OBQ) is asymptotically very similar to overlapping batch means, its performance for finite sample sizes is not. We show that the bias, the variance and the mean-squared-error of OBQ are not smooth functions of the batch size but rather cyclic. The cyclic behavior of OBQ depends on the marginal distribution, the point estimator of quantiles and the autocorrelation function and it diminishes with the sample size. We conclude that very large sample sizes and batch sizes are needed to obtain reliable standard error estimators when using OBQ, even for independently and identically distributed data.