Q-Q plots with confidence for testing Weibull and exponential distributions

Abstract

One of the basic graphical methods for assessing the validity of a distributional assumption is the Q-Q plot which compares quantiles of a sample against the quantiles of the distribution. In this paper, we focus on how a Q-Q plot can be augmented by intervals for all the points so that, if the population distribution is Weibull or exponential then all the points should fall inside the corresponding intervals simultaneously with probability 1 − α. These simultaneous 1 − α probability intervals provide therefore an objective mean to judge whether the plotted points fall close to the straight line: the plotted points fall close to the straight line if and only if all the points fall within the corresponding intervals. The powers of five Q-Q plot based graphical tests and the most popular non-graphical Anderson-Darling and Cramér-von-Mises tests are compared by simulation. Based on this power study, the tests that have better powers are identified and recommendations are given on which graphical tests should be used in what circumstances. Examples are provided to illustrate the methods.