A Control Chart to Monitor a Multivariate Binomial Process Una carta di controllo per monitorare un processo binomiale multivariato Paolo Cozzucoli 1 , Salvatore Ingrassia 2 1 Dipartimento di Economia e Statistica, Università della Calabria 2 Dipartimento di Economia e Metodi Quantitativi, Università di Catania e-mail: cozzucoli@unical.it Keywords: multivariate binomial process, two-sided control chart, ARL 1. Introduction 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 k- component multivariate binomial random vector with binomial marginal distribution X i ~B(n,p i ) and dependence structure specified by D and we write X~MVB k (p,n,D) with p=(p 1 ,…,p k ). The multivariate binomial distribution arises as follows: let Y 1 ,Y 2 ,…,Y n be iid multivariate Bernoulli vectors Y~MVB k (p,1,D), where D denotes a particular probability distribution on the 2 k binary k-tuples subject to the constraint that E(Y i )=p; then ∑ = = n i i 1 Y X ~ ) , , ( MVB D n k p , see e.g. Westfall and Young (1989). 2. An Index and a Control Chart for the Overall Defectiveness Let ( ) k i C C C ,..., ,..., 1 = C be the vector of k ordered quality defects. 1 C is the minimum unserious defect and k C is the maximum serious defect. Since different defects bring to the process different losses of quality then we may define a vector of weights that are numerical evaluations of the defectiveness degree of the product. In this case the sampling procedure, based on a sample of n items, determines a vector ( ) k i X X X ,..., ,..., 1 = X of random variables associated with the ( ) k i p p p ,..., ,..., 1 = p probability vector. Specifically, i X is the number of items in the sample with the i C defect and i p is the probability of observing in the selected item the i C defect. Consequently, the multivariate random variable X has a multivariate binomial distribution with parameters (n, p). Let ( ) k i c c c ,..., ,..., 1 = c be a vector of increasing weights associated to the C vector, where, 1 + < i i c c . In general, i c indicates the degree of quality loss that the th i − defect introduces into the system. These weights may be