Quantile Estimation in Dependent Sequences

Abstract

Standard nonparametric estimators of quantiles based on order statistics can be used not only when the data are i.i.d., but also when the data are drawn from a stationary, ϕ-mixing process of continuous random variables. However, when the random variables are highly positively correlated, the sample sizes needed for estimating extreme quantiles become computationally unmanageable. This paper gives a practical scheme, based on a maximum transformation in a two-way layout of the data, that reduces the sample size sufficiently to allow an experimenter to obtain a point estimate of an extreme quantile. The paper gives three schemes that lead to confidence interval estimates for the quantile. One uses a spectral analysis of the reduced sample. The other two, averaged group quantiles and nested group quantiles, are extensions of the method of batched means to quantile estimation. These two schemes give even greater data compaction than the first scheme. None of the schemes requires that the process being simulated is regenerative.