Estimation of the variance of percentile estimates

Abstract

The asymptotic formula for the variance of a percentile estimate is inversely proportional to the square of the probability density function evaluated at that percentile. In this note we show, for small and moderate sample sizes, that the estimate of the variance can have a moderate to large coefficient of variation even when the form of the density is known. When the density must be estimated empirically, the coefficient of variation increases substantially. We conclude that the estimate of the variance should not be used in either confidence interval estimation or hypothesis testing except for very large sample sizes.