A Control Chart to Monitor a Multivariate Binomial Process

The most applied statistical methods for monitoring multivariate attribute processes have been developed assuming that they have a multinomial distribution, see e.g. Marcucci (1985) and Cassady and Nachlas (2006). However this assumption is not always reasonable; indeed, it is more general and correct to suppose that in each item it is possible to identify one or more of k ordered and not mutually exclusive quality defects. In this case, the appropriate probabilistic model to monitor the process is the multivariate binomial distribution. Specifically, the sampling data may be modeled as coming from multivariate binomial distributions; let ( ) k i X X X ,..., ,..., 1 = X be a kcomponent multivariate binomial random vector with binomial marginal distribution Xi~B(n,pi) and dependence structure specified by D and we write X~MVBk(p,n,D) with p=(p1,...,pk). The multivariate binomial distribution arises as follows: let Y1,Y2,...,Yn be iid multivariate Bernoulli vectors Y~MVBk(p,1,D), where D denotes a particular probability distribution on the 2 binary k-tuples subject to the constraint that E(Yi)=p;