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

Abstract

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 is greater, equal and less than the determined value , respectively. Similar result holds for the Weibull distribution with and extends the one of two-parameter model. The Newton–Raphson method is used to determine the approximative value of the log-logistic model; it can thus lead to the reference shifted-exponential model, as for of the Weibull one. The relative variation between Weibull and log-logistic is also mentioned. Finally, two illustrative applications are provided.