Non-Parametric Estimation II. Statistically Equivalent Blocks and Tolerance Regions--The Continuous Case

Abstract

Wald [2, 1943] extended the usefulness of tolerance limits to the simplest multi-dimensional cases. His principle is here used to provide many new ways of using a sample of $n$ to divide the range of the population into $n + 1$ blocks of known behavior. The exact tolerance distribution for the proportions of the population covered by these blocks is extended from the case of a continuous probability density function to the case of a continuous cumulative distribution function. Such an extension is needed in dealing completely with multivariate cases even where the underlying distribution is as smooth as a multivariate normal distribution. The devices used in Paper I [1] to extend the usefulness of tolerance limits to the case of a discontinuous underlying distribution will be applied in the next paper of this series, with some extension, to extend the usefulness of these general tolerance regions to the case of a discontinuous distribution. Some of these results specialize into new results for the univariate case, although they do not seem to have any immediate practical application. The author wishes to acknowledge the stimulation given to his work on this problem by Henry Scheffe, whose modesty has kept this paper from the joint authorship of papers I [1, Scheffe and Tukey 1945] and IV (not yet written).