Properties and Applications of some Distributions derived from Frullani's integral

TL;DR

This paper reveals a probabilistic interpretation of Frullani's integral, introduces a new class of long-tailed distributions with finite moments, and demonstrates their applications in various fields through data fitting.

Contribution

It provides the first probabilistic interpretation of Frullani's integral and develops new long-tailed distributions with practical applications.

Findings

Distributions can have bimodal hazard functions.

Derived multivariate and skewed versions of the distributions.

Successfully fitted the distributions to real datasets.

Abstract

Frullani's integral dates from 1821, but a probabilistic interpretation of it has never been made. In this paper, Frullani's integral formula is shown to result from mixing a lifetime distribution by allowing the logarithm of the scale factor to be uniformly distributed over a finite range. This gives a class of long-tailed distributions related to slash distributions, where the pdf is simply expressed in terms of the survival function of the `parent' distribution. The resulting survival distributions have all moments finite, and can exhibit the bimodal hazard functions sometimes seen in practice. A distribution of this type analogous to the t-distribution is derived, the corresponding multivariate distributions are given, and two skewed versions of this distribution are derived. The use of the mixed distributions for inference is exemplified by fitting them to several datasets. It is expected that there will be many applications, in health, reliability, telecommunications, finance, etc.