Fisher dispersion index for multivariate count distributions: A review and a new proposal

The Fisher dispersion index is very widely used to measure the departure of any univariate count distribution from the equidispersed Poisson model. A multivariate extension has not yet been well defined and discussed in the literature. In this paper, a new definition of the multivariate Fisher index through the generalized dispersion index is proposed. This is a scalar quantity, defined as a ratio of two quadratic forms of the mean vector and the covariance matrix. A multiple marginal dispersion index and its relative extension for a given reference count distribution are discussed, and the asymptotic behavior and other properties are studied. Illustrative examples and practical applications on count datasets are analyzed under several scenarios. Some concluding remarks are made, including challenging problems.