A New Type 1 Alpha Power Family of Distributions and Modeling Data with Correlation, Overdispersion, and Zero-Inflation in the Health Data Sets

In the recent era, the introduction of a new family of distributions has gotten great attention due to the curbs of the classical univariate distributions. This study introduces a novel family of distributions called a new type 1 alpha power family of distributions. Based on the novel family, a special model called a new type 1 alpha power Weibull model is studied in depth. The new model has very interesting patterns and it is very flexible. Thus, it can model the real data with the failure rate patterns of increasing, decreasing, parabola-down, and bathtub. Its applicability is studied by applying it to the health sector data, and time-to-recovery of breast cancer patients, and its performance is compared to seven well-known models. Based on the model comparison, it is the best model to fit the health-related data with no exceptional features. Furthermore, the popular models for the data with exceptional features such as correlation, overdispersion, and zero-inflation in aggregate are explored with applications to epileptic seizer data. Sometimes, these features are beyond the probability distribution models. Hence, this study has implemented eight possible models separately to these data and they are compared based on the standard techniques. Accordingly, the zero-inflated Poisson-normal-gamma model which includes the random effects in the linear predictor to handle the three features simultaneously has shown its supremacy over the others and is the best model to fit the health-related data with these features.