The difference of symmetric quantiles under long range dependence

This paper investigates two robust estimators of the scale parameter given data from a stationary, long range dependent Gaussian process. In particular the limiting distributions of the interquartile range and related τ-quantile range statistics are established. In contrast to single quantiles, the limiting distribution of the difference of two symmetric quantiles is determined by the level of dependence in the underlying process. It is shown that there is no loss of asymptotic efficiency for the τ-quantile range relative to the standard deviation under extreme long range dependence which is consistent with results found previously for other estimators of scale.