Narrow Results

Granularity and perfect balance are defined and discussed for multiple factor designs. The granularity of a design is related to its discrepancy, an important concept in uniform experimental design. It indicates how fine a structure in the dependence of the response on the factors can be resolved. The balance of a design is similar to the resolution of fractional factorial designs, but it is defined for a much broader class of designs. The granularities and balance of various designs, including simple random designs, orthogonal arrays, digital nets, and integration lattices are compared. Two applications, the simple pendulum and blood glucose monitoring, are used to illustrate how granularity and balance can identify good designs.

By using the theory of spherical distributions and some properties of invariant statistics, we develop a stabilized uniform Q-Q plot for checking the multi-normality assumption in high-dimensional data analysis. Acceptance regions associated with the plot are given. Empirical performance of the acceptance regions is studied by Monte Carlo simulation. Application of the Q-Q plot is illustrated by a real data set.