Simultaneous Concentration Bands for Continuous Random Samples

Statisticians routinely plot ordered observations against expected values to validate a model using a random sample. In fact, it is possible to construct a probability plot for a random sample from any continuous distribution function, and this is accommodated by the probability integral transform which facilitates a uniform probability plot of the ordered transformed observations against their expected values. Random variation in the plots is sometimes assessed using pointwise concentration bands. There are two problems with these plots. First, under a given distribution, certain points are much more variable than others. For example, when the distribution is normal, the points nearest the middle of the plot have the smallest variance. Second, the order statistics used in the construction of the plots are correlated. Both problems make the interpretation of the plot difficult. Pointwise concentration bands are, however, inadequate because there will be departures from the expected 45◦ straight line not only from sampling variation but also from the correlation introduced by ordering the observations. To account for this correlation, we construct simultaneous concentration bands which have exact coverage probability. A comparison is made with the pointwise and Bonferroni concentration bands. An empirical study shows that it is beneficial to construct our exact simultaneous concentration bands, and reasonable departures from the underlying distribution assumption can be detected.