Abstract
This research investigates the use of a multidimensional range statistic as a tool to make statistical decisions concerning the variability of a process. Transformations were developed to permit the joint distribution of any combination of range and quasi-ranges (symmetric interior ranges) from dimension 1 to dimension m [less than or equal to] [n/2]. The transformations will accommodate any parent distribution and will permit the joint distribution of range and quasi-ranges to be expressed as a closed form integral expression when the density function and distribution function of the parent distribution are able to be expressed in closed form. The behavior of the uniform distribution was examined for a two-dimensional range statistic (outer and first-quasi-range) for a test of hypothesis concerning the variance. This was done for a sample size of four using three different classes of acceptance region boundaries, linear, second order and exponential. The sample size was increased to five and ten and the tests were redeveloped for the two higher order boundaries. In all cases the most powerful test was found to be equivalent to that which considered the outer range only, a counterintuitive result.
Loomis, Robert James (1985). The development and analysis of multidimensional range statistics. Texas A&M University. Texas A&M University. Libraries. Available electronically from
https : / /hdl .handle .net /1969 .1 /DISSERTATIONS -597607.