Simultaneous Confidence Intervals for Several Quantiles of an Unknown Distribution

Given a sample of independent observations from an unknown continuous distribution, it is standard practice to construct a confidence interval for a specified quantile of the distribution using the binomial distribution. Furthermore, confidence bands for the unknown cumulative distribution function, such as Kolmogorov’s, provide simultaneous confidence intervals for all quantiles of the distribution, which are necessarily wider than the individual confidence intervals at the same confidence level. The purpose of this article is to show how simultaneous confidence intervals for several specified quantiles of the unknown distribution can be calculated using probabilities from a multinomial distribution. An efficient recursive algorithm is described for these calculations. An experimenter may typically be interested in several quantiles of the distribution, such as the median, quartiles, and upper and lower tail quantiles, and this methodology provides a bridge between the confidence intervals with individual confidence levels and those that can be obtained from confidence bands. Some examples of the implementation of this nonparametric methodology are provided, and some comparisons are made with some parametric approaches to the problem.