The exponential variation property for the Weibull distribution

In this paper, our aim is to characterize the Weibull distribution widely used to model lifetimes in reliability analysis. Namely, for the exponential variation index defined as the squared of the coefficient of variation, we show that the twoparameter Weibull distribution is over-, equi- and under-varied if and only if its shape parameter is less, equal and greater than one, respectively. In reliability, similar behaviours on the shape parameter are known for the decay, constant and growth of the Weibull failure rate curve. To this end, we have stated the main result which is analytically proved by means of sharp inequalities involving the polygamma functions.