Linear Combination of Independent Exponential Random Variables

Li, Kim-Hung; Li, Cheuk Ting

doi:10.1007/s11009-018-9653-0

Cited by 10 publications

(13 citation statements)

References 32 publications

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“…. , n are not all identical, is called (general) hypoexponential distribution (see [1,2]). It is absolutely continuous and we denote by g n its density.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Exponential and Hypoexponential Distributions: Some Characterizations

Yanev

2020

Mathematics

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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.

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“…. , n are not all identical, is called (general) hypoexponential distribution (see [1,2]). It is absolutely continuous and we denote by g n its density.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The obtained characterization seems of interest on its own, but it can also serve as a basis for further investigations of intermediate cases of mixed type with some ties and at least two distinct parameters (see [2]). Of certain interest is also the case where not all weights µ i 's are positive (see [1]).…”

Section: Discussionmentioning

confidence: 99%

Exponential and Hypoexponential Distributions: Some Characterizations

Yanev

2020

Mathematics

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“…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.…”

Section: Individual Perception and Consumptionmentioning

confidence: 99%

“…( 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.…”

Section: Micro Level Patternsmentioning

confidence: 99%

A Network Approach to Consumption

Schulz¹,

Mayerhoffer²

2021

SSRN Journal

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The nexus between debt and inequality has attracted considerable scholarly attention in the wake of the global financial crisis. One prominent candidate to explain the striking co-evolution of income inequality and private debt in this period has been the theory of upward-looking consumption externalities leading to expenditure cascades. We propose a parsimonious model of upward-looking consumption at the micro level mediated by perception networks with empirically plausible topologies. This allows us to make sense of the ambiguous empirical literature on the relevance of this channel. Up to our knowledge, our approach is the first to make the reference group to which conspicuous consumption relates explicit. Our model, based purely on current income, replicates the major stylised facts regarding micro consumption behaviour and is thus observationally equivalent to the workhorse permanent income hypothesis, without facing its dual problem of 'excess smoothness' and 'excess sensitivity'. We also demonstrate that the network topology and segregation has a significant effect on consumption patterns which has so far been neglected.

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“…Under the assumption that the rate parameters λ i > 0 are pairwise distinct, the distribution of the sum S N = i∈N X i can be represented as a generalized exponential mixture (GEM) with distribution function F S N (x) = 1 − i∈N π i e −λ i x , x > 0, with real-valued mixing proportions π i which satisfy i∈N π i = 1. Note that this distribution class is also known in the literature under other names, such as generalized Erlang [9], hypoexponential, as its coefficient of variation is smaller than with the exponential distribution [19], or generalized hyperexponential [13].…”

Section: Introductionmentioning

confidence: 99%

Explicit results on conditional distributions of generalized exponential mixtures

Klüppelberg

Seifert

2020

J. Appl. Probab.

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For independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

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Linear Combination of Independent Exponential Random Variables

Cited by 10 publications

References 32 publications

Exponential and Hypoexponential Distributions: Some Characterizations

Exponential and Hypoexponential Distributions: Some Characterizations

A Network Approach to Consumption

Explicit results on conditional distributions of generalized exponential mixtures

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