Inflated distributions are applied in various fields, including insurance, traffic networks, and survival analyses. First, they are defined by a baseline discrete distribution, and then extra masses are added to some points of interest, called inflated points, to achieve more flexible models for data analyses. The baseline distribution is arbitrary and application-dependent. Here, the rich family of power series distributions is considered as the baseline, which includes various common discrete distributions such as Poisson, negative binomial, multinomial, and logarithmic series distributions. This paper deals with an extension of previous works in two directions. The former is an extension of the univariate inflated distributions to multivariate ones, and the latter is the generalization of the inflated points from the zero single point to the set k = 0, 1, .... Under this setting, various introduced inflated distributions in the literature fall into the proposed family of distributions. These extensions make the proposed model flexible and practically useful in data analyses. To do this, the problem of estimating parameters with various approaches as well as hypotheses testing is studied in detail. Multivariate generalized linear models with inflated multivariate discrete responses are also discussed. To assess the performance of the proposed family of inflated distributions, simulation studies are conducted, and a real data set on an Australian health survey study is also analyzed.

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