Procedures for Separations within Batches of Values, I. The Orderly Tool Kit and Some Heuristics

摘要

First of several reports studying techniques to answer: Is it desirable to treat this batch of values together, or to separate them into at least two parts (called below the canyon problem). This report focuses on tools to help answer the question, primarily on the orderly tool kit in which the simple general properties of order statistics from any reasonable distribution are identified and arranged for use. Once we order the values of a set of observations, our attention needs to shift from values to gaps differences of adjacent ordered values. Rescaling gaps toward a common scale allows easy combination into measures of spread. Like other measures of spread, these measures of spread are well described in terms of degrees of freedom, which can be defined for both linear and quadratic expressions. (Linear d.f. will be slightly less than 4 times quadratic d.f.) It can be natural and useful to order the rescaled gaps, when we now deal with gaps of rescaled gaps. (Various quantitative matters are also addressed.) Application of the orderly tool kit to the canyon problem leads to test statistics for whose distribution a well-chosen F-distribution is a natural approximation. While the orderly tool kit provides a rough idea of the power of the procedure, simulation will be required to anchor and modify the approximation. (KR)