Exponential and Hypoexponential Distributions: Some Characterizations

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≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’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 Villaseñor (2013) for a particular convolution of two random variables.

On a characterization of exponential and double exponential distributions

“…The proof of the theorem is given in Section 2. Theorem 1.1 is an extension of a similar result of G. Yanev [7] obtained under the additional assumption that the coefficients µ k are positive. In that case, the key technical condition (3) is trivial as the left-hand sides contains the µ m k terms and hence is always larger than the right-hand side.…”

“…In the particular case when X ∈ E and µ k = 1 L−k+1 for some integer L > n, the random variable S/λ is distributed as the n-th order statistic of a sample of L independent copies of X (this is the Rényi representation of order statistics; see, for instance, [3, p. 18]). For further background and earlier versions (particular cases) of Yanev's characterization theorem see [1,6,7].…”

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Recently, G. Yanev [7] obtained a characterization of the exponential family of distributions in terms of a functional equation for certain mixture densities. The purpose of this note is twofold: we extend Yanev's theorem by relaxing a restriction on the sign of mixture coefficients and, in addition, obtain a similar characterization for the Laplace family of distributions.

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A Network Approach to Consumption

“…While more distant incomes are thus generally higher due to social comparisons being upward-looking, they are also given less weight. This implies that C(i) is a linear combination of an exponentially distributed random variable Y with varying weights and number of terms and thus, a so-called hypoexponential mixture (Li & Li 2019;Yanev 2020). As we discuss in more detail in section 4, this directly implies two of the four stylised facts, namely, the approximate log-normality of expenditure distributions and the fact that they are robustly more homogeneous than income distributions.…”

“…( 3), expenditure levels are a weighted sum of income levels, themselves following an exponential distribution, with varying weights and number of terms. This characteristic implies that the distribution of C is hypoexponential, with a coefficient of variation strictly smaller than unity, while the exponentially distributed income levels exhibit a coefficient of variation (asymptotically) equal to unity (Li & Li 2019;Yanev 2020). Intuitively, since social consumption accumulates in a cascade down the income distribution, richer individuals are relatively unaffected by status concerns, while the cumulative effect on poor individuals is much higher.…”