ABSTRACT
A different approach can be taken to determine whether an ob servation, or a group of observations, come from a given multivariate normal distribution. Instead of setting up a “null hypothesis,” one can identify a region which, under statistical control, contains a certain pro portion of the multivariate observations. When the parameters μ and Σ of the distribution are known and the resulting region is called a normal process region, already mentioned in Chapter 1. Each new observation Y is tested to determine whether it belongs to the prespecified region by computing
which is compared to the appropriate percentiles of the chi-squared dis tribution with p degrees of freedom. We illustrate the normal process regions with the data from Case Study 1. Assuming a multivariate nor mal distribution with parameters equal to the empirical estimates from the reference sample of size 30, we use the ungrouped estimate for the covariance matrix to compute the value of T2(Y) for the 66th observation whose measurements on the six dimensions are
The value of T2(Y) = 82.63 is larger than the 99.999-th percentile of the chi-squared distribution with 6 degrees of freedom. We thus conclude that the new observation does not belong to the mid 99.999% of the population defined by the “reference sample.”
