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Summary: Multivariate zero‐inflated Poisson (ZIP) distributions are important tools for modelling and analysing correlated count data with extra zeros. Unfortunately, existing multivariate ZIP distributions consider only the overall zero‐inflation while the component zero‐inflation is not well addressed. This paper proposes a flexible multivariate ZIP distribution, called the multivariate component ZIP distribution, in which both the overall and component zero‐inflations are taken into account. Likelihood‐based inference procedures including the calculation of maximum likelihood estimates of parameters in the model without and with covariates are provided. Simulation studies indicate that the performance of the proposed methods on the multivariate component ZIP model is satisfactory. The Australia health care utilisation data set is analysed to demonstrate that the new distribution is more appropriate than the existing multivariate ZIP distributions. [ABSTRACT FROM AUTHOR]