Abstract
This study on the Type-I heavy-tailed Rayleigh (TI-HTR) distribution is a special case of Type-I heavy-tailed (TI-HT) family of distributions was studied. The characteristics were derived, including the moment and its measures, quantile function, reliability measures, and other statistical properties as well as parameter estimation using the maximum likelihood method and penalized likelihood estimation. The behavior of its various functions were shown graphically. Analytically, we showed that model linearly grows near the origin and exhibits rapid exponential decay. However, the tail behavior cannot equal the traditional heavy-tail in the power law sense, hence it is called the type-I heavy-tail. Interestingly, we designed a group acceptance plan (GASP) and demonstrated usefulness with both assumed and maximum likelihood estimates. The GASP under the TI-HTR distribution is preferable when the parameter values are small. The distribution was used to model real-life data sets to justify its usefulness. The results of the application of the model to both COVID-19 and Cancer data showed that the model fits the two data better than the competing models and also suggest that inference from the model is better than those of the competitors. In estimating the parameters, the penalized likelihood procedure perform considerably better with minimum standard error of the estimates. From the Cramér-von Mises test results which guides against the heavy-tail sensitivity, the TI-HTR distribution offers a better model for fitting fast decaying exponential data since it has the least statistics in both datasets.