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Confidence Bands for Quantile Functions: A Parametric and Graphic Alternative for Testing Goodness of Fit

Abstract This article derives simultaneous 100(1 − α)% confidence intervals for the quantiles of a normal distribution using a method first proposed by Cheng and Iles and independently rediscovered by Satten. These methods yield a novel parametric/graphic alternative to the usual goodness-of-fit tests for normality based on normal Q-Q plots, or extensions of Kolmogorov's test based on the empirical distribution function: Draw the graph of the empirical quantile plot, then plot the corresponding upper and lower bounds for these quantiles. We thus obtain a 100(1 − α)% confidence band for the empirical quantile plot. The more the parent distribution of the data departs from the normal, the more likely the empirical quantile plot will have points that lie outside these bounds. The empirical quantile plot and its confidence bands are pleasing to the eye and easy to interpret, even for a user unfamiliar with the underlying theory. Our bounds, although not optimal, are easily computable in terms of the normal, t, and χ2 distributions. The test is applicable for all sample sizes n (no delicate asymptotic limit theorems are required) and it works when the mean and variance are estimated from the data. These methods are illustrated by applying them to two datasets analyzed by different methods.