scholarly journals Concentration inequalities for polynomials in α-sub-exponential random variables

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Friedrich Götze ◽  
Holger Sambale ◽  
Arthur Sinulis
Keyword(s):  
Random Variables ◽  
Concentration Inequalities ◽  
Exponential Random Variables

Testing for a change point in a sequence of exponential random variables with repeated values

2010 ◽  
Vol 80 (2) ◽  
pp. 191-199 ◽  
Author(s):  
Asoka Ramanayake ◽  
Arjun K. Gupta
Keyword(s):  
Change Point ◽  
Random Variables ◽  
Exponential Random Variables

Order Statistics of Bivariate Exponential Random Variables

1996 ◽  
pp. 129-141 ◽  
Author(s):  
H. N. Nagaraja ◽  
Geraldine E. Baggs
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Bivariate Exponential ◽  
Exponential Random Variables

scholarly journals Distribution of Minimal Path Lengths when Edge Lengths are Independent Heterogeneous Exponential Random Variables

2012 ◽  
Vol 49 (3) ◽  
pp. 895-900
Author(s):  
Sheldon M. Ross
Keyword(s):  
Joint Distribution ◽  
Shortest Paths ◽  
Random Variables ◽  
Simulated Data ◽  
Minimal Path ◽  
Exponential Random Variables ◽  
Path Lengths

We find the joint distribution of the lengths of the shortest paths from a specified node to all other nodes in a network in which the edge lengths are assumed to be independent heterogeneous exponential random variables. We also give an efficient way to simulate these lengths that requires only one generated exponential per node, as well as efficient procedures to use the simulated data to estimate quantities of the joint distribution.

Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk

2017 ◽  
Vol 49 (2) ◽  
pp. 481-514 ◽  
Author(s):  
Yunpeng Sun ◽  
Rafael Mendoza-Arriaga ◽  
Vadim Linetsky
Keyword(s):  
Credit Risk ◽  
Random Variables ◽  
Exponential Distributions ◽  
Efficient Simulation ◽  
Failure Times ◽  
Independent Unit ◽  
Exponential Random Variables ◽  
Credit Portfolio ◽  
Multivariate Exponential ◽  
Passage Times

Abstract In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

scholarly journals Erlang Strength Model for Exponential Effects

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Gökhan Gökdere ◽  
Mehmet Gürcan
Keyword(s):  
Finite Number ◽  
System Performance ◽  
Random Variables ◽  
State System ◽  
Strength Model ◽  
Component System ◽  
Exponential Random Variables ◽  
Complete Failure ◽  
Technical Systems

AbstractAll technical systems have been designed to perform their intended tasks in a specific ambient. Some systems can perform their tasks in a variety of distinctive levels. A system that can have a finite number of performance rates is called a multi-state system. Generally multi-state system is consisted of components that they also can be multi-state. The performance rates of components constituting a system can also vary as a result of their deterioration or in consequence of variable environmental conditions. Components failures can lead to the degradation of the entire multi-state system performance. The performance rates of the components can range from perfect functioning up to complete failure. The quality of the system is completely determined by components. In this article, a possible state for the single component system, where component is subject to two stresses, is considered under stress-strength model which makes the component multi-state. The probabilities of component are studied when strength of the component is Erlang random variables and the stresses are independent exponential random variables. Also, the probabilities of component are considered when the stresses are dependent exponential random variables.

COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Exponential Random Variables ◽  
Binomial Random Variables

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

On the distribution of the sum of independent and non-identically extended exponential random variables

2019 ◽  
Vol 3 (1) ◽  
pp. 23-37
Author(s):  
Masato Kitani ◽  
Hidetoshi Murakami
Keyword(s):  
Random Variables ◽  
Exponential Random Variables

Classes of probability density functions having Laplace transforms with negative zeros and poles

1987 ◽  
Vol 19 (3) ◽  
pp. 632-651 ◽  
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Probability Density ◽  
Practical Importance ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Laplace Transforms ◽  
Probability Density Functions ◽  
Independent Random Variables ◽  
Density Functions ◽  
Exponential Random Variables ◽  
Zeros And Poles

We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+ of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency of f ∊ Ω to be in Ω+. The class Ω+ contains many classes of great importance such as mixtures of n independent exponential random variables (CMn), sums of n independent exponential random variables (PF∗n), sums of two independent random variables, one in CMr and the other in PF∗1 (CMPFn with n = r + l) and sums of independent random variables in CMn(SCM). Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.

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