Approximating the Sampling Distribution of the T2 Statistic

Introduction The sampling distribution of a T statistic is well established when the statistic is based on observations taken from a multivariate normal population (MVN). With known parameters, the sampling distribution is always a chi-square distribution. In a Phase I operation with unknown parameters, where outliers need to be detected, the distribution of the T statistic is a beta distribution. In a Phase II operation with unknown parameters, where new observations are being monitored, the sampling distribution is an F distribution. When the data distribution is not an MVN, we know of no existing procedures for determining the sampling distribution. The objective of this paper is to present a methodology for using the known distributions of the T statistic under the MVN assumption as approximations to the sampling distribution of the T statistic based on process data obtained from a non-normal multivariate distribution. Our procedure is based on the premise that if two distributions have a finite number of lower moments in common, the distributions often will be approximately the same (see Kendall and Stuart 1967). We examine the lower two moments of the T statistic under the assumption that X ~ Np(μ,∑) (1) and compare them to the lower two moments of the T statistic when X ~ MVp(μ,∑). (2) Conditions are established as to when the moments of the T statistic under (1) and (2) agree, so that the known distributions may be used as approximations to the unknown sampling distribution of the T statistic. Moments of T with Known Parameters Consider a sample where X' = (x1,x2,..,xp) and condition (1) is satisfied with known parameters μ and ∑. The distribution of the T statistic for this data is a chi-square distribution with p degrees of freedom, T = (X-μ)'∑(X-μ) ~ χp. (3) Thus, the moments of the T distribution are exactly equal to the moments of the chi-square distribution in (3). For example, the first moment of the T distribution is given by E(T) = p. If the parameters are known but the distribution of X is unknown as in case (2), the moments of the T statistic can be derived using the MYT decomposition (Mason, et al (2002)). Two types of terms are contained in a MYT decomposition: unconditional and conditional. The general form of the unconditional term is given as