Exponential and Hypoexponential Distributions: Some Characterizations

• Main Author: George P. Yanev
• Format: Article
• Language: English
• Published: MDPI AG 2020-12-01
• Series: Mathematics
• Subjects: exponential distribution; hypoexponential distribution; characterizations
• Online Access: https://www.mdpi.com/2227-7390/8/12/2207

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 <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mspace width="0.166667em"></mspace><mi>…</mi><mo>,</mo><mspace width="0.166667em"></mspace><msub><mi>X</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are independent copies of a random variable <i>X</i> with unknown distribution <i>F</i> and a specific linear combination of <inline-formula><math display="inline"><semantics><msub><mi>X</mi><mi>j</mi></msub></semantics></math></inline-formula>’s has hypoexponential distribution, then <i>F</i> 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 Villaseñor (2013) for a particular convolution of two random variables.