The Yun transform in probabilistic and statistical contexts: Weibull baseline case and its applications in reliability theory

Abstract In this paper, we present a new family of distributions based on a particular case of a transform introduced by Yun (2014). Among others, this transform demonstrates great flexibility and nice mathematical properties which can be useful in a statistical context (continuous derivatives of all order, simplicity of the inverse transform, etc.). We propose a new three-parameter distribution from this family, namely the Yun–Weibull (YW) distribution. Some statistical properties of this distribution are studied, involving flexible hazard rate shapes. Subsequently, the statistical inference of the YW distribution is investigated. The parameters are estimated by employing the maximum likelihood estimation method. We establish the existence and uniqueness of the obtained estimators. The YW distribution is applied to fit two practical data sets. As a main result of our analysis, the new distribution is found to be more appropriate to these data sets than other competitive distributions. Moreover, the uniqueness of the parameter estimates of the YW distribution is studied using the profile log-likelihood function visually under the two practical data sets.