Distribution-Free Confidence Intervals

Abstract

In recent years, more and more statisticians have come to appreciate the advantages of nonparametric tests. Not only do nonparametric tests have often surprisingly high efficiency relative to their normaltheory equivalents even under assumption of normality, but they are also less sensitive to the influence of wild observations than are the normal-theory equivalents. Much less appreciated seems to be the fact that many nonparametric tests can also be converted into confidence intervals for suitably chosen parameters. The basic idea is very simple. A nonparametric null hypothesis is formulated in such a way that it involves the parameter for which we want to find a confidence interval. The confidence interval then consists of the set of parameter values for which the null hypothesis is not rejected. To the extent that the test procedure is distribution-free, the resulting confidence interval is also distribution-free. In general, trial and error methods are required to determine the boundaries between acceptable and nonacceptable parameter values. Often it is possible to systematize this trial and error approach. And in some cases we can actually specify the endpoints in terms of the available observations. In what follows we shall investigate a class of procedures of this type. In general, the results are not new, though no attempt will be made to provide original references. When using nonparametric procedures it is customary to assume that sampled populations are continuous in order to insure the distribution-free character of the procedure when the null hypothesis is true. For the present we shall make the same assumption, though we shall see later that omission of this assumption does not cause any undue difficulties as far as confidence intervals are concerned. We illustrate our approach with a very simple and well-known example. Let us find a confidence interval for the median -q of an arbitrary (continuous) population on the basis of a random sample z1, . . ., Zn. We start with the sign test of the hypothesis Ho: -q = mqo against a two-sided alternative. The test can be performed as follows. We define two statistics,