Exponential and Hypoexponential Distributions: Some Characterizations

TL;DR

This paper characterizes the exponential distribution by showing that certain linear combinations of independent variables with hypoexponential distributions imply the original variables are exponential, extending previous characterizations.

Contribution

It proves a new converse characterization of the exponential distribution based on hypoexponential sums with distinct rates.

Findings

Linear combinations of independent variables with hypoexponential distributions imply the variables are exponential.

Extends previous characterizations of exponential distributions for specific convolutions.

Provides new insights into the structure of hypoexponential and exponential distributions.

Abstract

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some $n\ge 2$, $X_1, X_2,\,\ldots,\,X_n$ are independent copies of a random variable $X$ with unknown distribution $F$ and a specific linear combination of $X_j$'s has hypoexponential distribution, then $F$ is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villase\~{n}or (2013) for a particular convolution of two random variables.