The Shifted-Exponential Variation Property for the Weibull and Log-Logistic Models

In this paper, the recent shifted-exponential variation property which is defined as the ratio of variance to the squared of shifted expectation is investigated for both three-parameter Weibull and log-logistic models. These nonnegative semicontinuous models are widely considered in engineering, economics, hydrology, demography and many other fields. It is shown that the log-logistic distribution corresponds to over-, equi-, and under-varied if and only if its only positive shape parameter $$\beta$$ is greater, equal and less than the determined value $$\beta_{1}\in(0,1/2)$$ , respectively. Similar result holds for the Weibull distribution with $$\beta_{1}=1$$ and extends the one of two-parameter model. The Newton–Raphson method is used to determine the approximative value $$\beta_{1}=0.37100965$$ of the log-logistic model; it can thus lead to the reference shifted-exponential model, as for $$\beta_{1}=1$$ of the Weibull one. The relative variation between Weibull and log-logistic is also mentioned. Finally, two illustrative applications are provided.