Bootstrapping Sample Quantiles of Discrete Data

Sample quantiles are consistent estimators for the true quantile and satisfy central limit theorems (CLTs) if the underlying distribution is continuous. If the distribution is discrete, the situation is much more delicate. In this case, sample quantiles are known to be not even consistent in general for the population quantiles. In a motivating example, we show that Efron’s bootstrap does not consistently mimic the distribution of sample quantiles even in the discrete independent and identically distributed (i.i.d.) data case. To overcome this bootstrap inconsistency, we provide two different and complementing strategies. In the first part of this paper, we prove that m-out-of-n-type bootstraps do consistently mimic the distribution of sample quantiles in the discrete data case. As the corresponding bootstrap confidence intervals tend to be conservative due to the discreteness of the true distribution, we propose randomization techniques to construct bootstrap confidence sets of asymptotically correct size. In the second part, we consider a continuous modification of the cumulative distribution function and make use of mid-quantiles studied in Ma, Genton and Parzen (2011). Contrary to ordinary quantiles and due to continuity, mid-quantiles lose their discrete nature and can be estimated consistently. Moreover, Ma, Genton and Parzen (2011) proved (non-)central limit theorems for i.i.d. data, which we generalize to the time series case. However, as the mid-quantile function fails to be differentiable, classical i.i.d. or block bootstrap methods do not lead to completely satisfactory results and m-out-of-n variants are required here as well. The finite sample performances of both approaches are illustrated in a simulation study by comparing coverage rates of bootstrap confidence intervals. Introduction Since the seminal work of Efron (1979), bootstrapping has been established as a major tool for estimating unknown finite sample distributions of general statistics. Among others, this method has successfully been applied to construct confidence intervals for sample quantiles of continuous distributions; see e.g. Serfling (2002, Chapter 2.6), Sun and Lahiri (2006), Sharipov and Wendler (2013) and references therein. In this case, the asymptotic behavior of quantile estimators is well-understood. Based on the well-known Bahadur representation, a CLT can then be established for sample quantiles in case of an underlying distribution exhibiting a differentiable cumulative distribution function (cdf) and a positive density at the quantile level of interest. This allows for the application of classical results on the bootstrap to mimic the unknown finite sample distribution. If the underlying distribution is discrete, the situation is much more delicate. Sample quantiles may not even be consistent in general for the population quantiles in this case. This issue occurs due to the fact that the cdf is a step function. This leads to inconsistency if the level of Date: May 15, 2014. 1991 Mathematics Subject Classification. 62G09, 62M10, 62G05. JEL subject code C13, C15.