A new alpha power type-1 family of distributions and modelling the overdispersed count outcome

In this paper‎, ‎we introduce a novel family of statistical models called a new alpha power type-1 family of distributions‎. ‎Three sub-cases of the family are discussed‎. ‎Based on the novel family‎, ‎a special model‎, ‎explicitly‎, ‎a new alpha power type-1-Weibull distribution is studied in depth‎. ‎The new model has very interesting patterns of failure rates like increasing‎, ‎decreasing‎, ‎bathtub‎, ‎and parabola-down‎. ‎Hence‎, ‎it is so flexible‎. ‎Based on the comparison analysis‎, ‎among five well-known models‎, ‎it has an impact on health data analysis‎. ‎Furthermore‎, ‎the count data models capable of handling overdispersion and zero-inflation are discussed and applied the real health data‎. ‎The zero-inflated negative binomial model in the frequentist approach has shown its popularity in handling both overdispersion and zero-inflation simultaneously‎, ‎while the discrete Weibull model with the logit(q) link in the Bayesian approach outperformed its counterparts.