A Loglikelihood-Based Shape Measure of Past Lifetime Distribution

Abstract

Generalized past information measure based on Rényi entropy for a continuous random variable useful in life testing is considered and its connection to the loglikelihood is established. From this relation, a loglikelihood-based past life distribution measure ${\overline{S}}_f\left(t\right)$ has been proposed. From the gradient of the spectrum of generalized past information, the ${\overline{S}}_f\left(t\right)$ measure has been calculated for some important parametric life distributions. Serving as a measure of the shape of a past life (inactivity time) distribution, ${\overline{S}}_f\left(t\right)$ can be used to compare the tails and shapes of various frequently used densities, where the traditional kurtosis measure is not applicable.