A direct method for constructing distribution-free tolerance regions

Abstract

Distribution-free tolerance regions are hyper-rectangles in terms of the number of variables that include at least a pre-specified proportion of normal subjects with a confidence bounded from below by a prescribed probability. This paper has two main goals. The first is to propose an innovative method for constructing multidimensional tolerance regions that work well when the only assumption that can be made about the underlying distribution is that it is continuous. Although our proposal is, in essence, an extension of the Wilks-Wald method to higher dimensions, this research is far less immediate and straightforward than it has appeared to many authors, who, moreover, have not really used it in practical work. In particular, the order statistics dividing the regions are not fixed beforehand, but are determined by an optimization procedure. The second goal is to suggest a way of overcoming a problem of practical importance concerning the order in which the variables are included, which has remained unsolved since the introduction of the argument almost eighty years ago. This device facilitates the search for a good solution while avoiding being drawn away into the black hole of combinatorial computations. Simulation and applications to laboratory medicine data illustrate the advantages of using the method presented in this paper.

Distribution–free hyperrectangular tolerance regions for setting multivariate reference regions in laboratory medicine

“…The simulation study in Reference 37 on the true confidence levels of the hyperrectangular and upper‐semi tolerance regions of Young and Mathew 18 under various bivariate mixed normal and mixed $$$ t $$$ distributions shows that the true confidence levels could be considerably larger than $$$ \gamma $$$ sometimes. For example, when $$$ P=90\%,\gamma =0.90 $$$ and $$$ n=1000 $$$, the true confidence level of the hyperrectangular tolerance region (using simplicial depth) of Young and Mathew 18 is estimated to be 0.944 from simulation (Reference 36, p. 26, tab. 4.3).…”

“…The gist of our construction method is that the true confidence level can be computed exactly depending on the total number of EBs without specifying the multivariate probability model assumed to generate the observed data. Besides, as noted by Di Bucchianico et al 35 and Amerise, 36 there is no canonical ordering in higher dimensions thus auxiliary ordering functions are implied in Tukey's method. This means that one cannot look at the data and select the best ordering that results in the smallest hyperrectangle.…”