This research investigates the construction of regression models for scenarios in which the response variable is inflated at specific points. To address this, we propose a comprehensive family of inflated distributions, which encompasses virtually all standard inflated distributions as special cases. The proposed family of distributions is applicable when the variable of interest is discrete, continuous, or a combination of both. We discuss parameter estimation, develop a regression model using the introduced family of distributions, and formulate an expectation-maximization (EM) algorithm to determine the maximum likelihood estimators of the proposed regression model. Additionally, we develop a general likelihood ratio test for the regression parameters. Finally, in two simulation scenarios and two real data sets, (obtained from the US National Center for Health Statistics (NCHS) and the residents of Olmsted County aged 50 or older), we analyze the performance of the proposed model.

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