Confidence bands for distribution functions when parameters are estimated from the data: a non-Monte-Carlo approach.

A method is given for computing simultaneous confidence intervals for order statistics obtained from a distribution depending on one or more parameters that must be estimated from the data. This produces a confidence band for the distribution itself and may be regarded as an extension of Kolmogorov's goodness-of-fit test to the case where the distribution depends on parameters that must be estimated from the data. The method works whenever the joint confidence set for the parameters is convex and the quantile function is linear in the parameters. Two special cases are treated in some detail: the normal and exponential distributions. Graphical representations and comparisons with results obtained by Lillifors and Stephens via Monte-Carlo methods are discussed. An unusual feature of this paper is that we found it necessary to first prove that the joint confidence set for the mean and variance for the normal distribution based on the Wald statistic is convex and compact. Our proof relies on an elementary theorem from differential geometry in the large due to Hopf and is of independent interest.