A Bayes classifier when the class distributions come from a common multivariate normal distribution

Let (n-1) measurements be taken on a component of some manufactured product prior to the manufacture of the product. The author wants to decide whether to keep or reject this component before it is allowed to enter the manufacturing process, based on the relationship of these measurements to some postmanufacture measurement on the finished product. Let all measurements (`before´ and `after´), taken together, form an n-dimensional random vector described by a multivariate normal distribution. The mathematical relationships necessary for the design of a Bayes classifier for the component is derived. The classifier has a relatively simple form, and can be easily implemented on a personal computer. There are two situations where this new classifier is an attractive alternative to the traditional classifier: (1) the assumption of two distinct normal distributions for the good and bad classes is theoretically untenable, and (2) the estimation of two different covariance matrices would be difficult for economic or other practical reasons