Bivariate life distributions from Pólya's urn model for contagion

1993 ◽  
Vol 30 (03) ◽  
pp. 497-508 ◽  
Author(s):  
Albert W. Marshall ◽  
Ingram Olkin
Keyword(s):  
Poisson Processes ◽  
Urn Model ◽  
Exponential Distributions ◽  
Bivariate Exponential ◽  
Shock Models ◽  
The Family ◽  
Life Distributions ◽  
Pólya’S Urn ◽  
Special Case ◽  
Multivariate Exponential

Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

Bivariate life distributions from Pólya's urn model for contagion

1993 ◽  
Vol 30 (3) ◽  
pp. 497-508 ◽  
Author(s):  
Albert W. Marshall ◽  
Ingram Olkin
Keyword(s):  
Poisson Processes ◽  
Urn Model ◽  
Exponential Distributions ◽  
Bivariate Exponential ◽  
Shock Models ◽  
The Family ◽  
Life Distributions ◽  
Pólya’S Urn ◽  
Special Case ◽  
Multivariate Exponential

Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

scholarly journals Zipf’s, Heaps’ and Taylor’s Laws are Determined by the Expansion into the Adjacent Possible

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 752 ◽  
Author(s):  
Francesca Tria ◽  
Vittorio Loreto ◽  
Vito Servedio
Keyword(s):  
Ad Hoc ◽  
Ecological Systems ◽  
Dirichlet Processes ◽  
Urn Model ◽  
Modeling Framework ◽  
Self Consistent ◽  
Pólya’S Urn ◽  
Innovation Rate ◽  
Innovation Processes ◽  
Simple Modeling

Zipf’s, Heaps’ and Taylor’s laws are ubiquitous in many different systems where innovation processes are at play. Together, they represent a compelling set of stylized facts regarding the overall statistics, the innovation rate and the scaling of fluctuations for systems as diverse as written texts and cities, ecological systems and stock markets. Many modeling schemes have been proposed in literature to explain those laws, but only recently a modeling framework has been introduced that accounts for the emergence of those laws without deducing the emergence of one of the laws from the others or without ad hoc assumptions. This modeling framework is based on the concept of adjacent possible space and its key feature of being dynamically restructured while its boundaries get explored, i.e., conditional to the occurrence of novel events. Here, we illustrate this approach and show how this simple modeling framework, instantiated through a modified Pólya’s urn model, is able to reproduce Zipf’s, Heaps’ and Taylor’s laws within a unique self-consistent scheme. In addition, the same modeling scheme embraces other less common evolutionary laws (Hoppe’s model and Dirichlet processes) as particular cases.

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.

Reliability assessment of the standby system with dependent components by bivariate exponential distributions

2021 ◽  
pp. 1748006X2110414
Author(s):  
Afshin Yaghoubi ◽  
Peyman Gholami
Keyword(s):  
Reliability Function ◽  
Active Components ◽  
Exponential Distributions ◽  
Bivariate Exponential Distribution ◽  
Multiple Components ◽  
Proposed Model ◽  
Bivariate Exponential ◽  
Real World Applications ◽  
Model Dependency ◽  
Reduced Time

In the reliability analysis of systems, all system components are often assumed independent and failure of any component does not depend on any other component. One of the reasons for doing so is that considerations of calculation and elegance typically pull in simplicity. But in real-world applications, there are very complex systems with lots of subsystems and a choice of multiple components that may interact with each other. Therefore, components of the system can be affected by the occurrence of a failure in any of the components. The purpose of this paper is to give an explicit formula for the computation of the reliability of a system with two parallel active components and one spare component. It is assumed that parallel components are dependent and operate simultaneously. Two distributions of Freund’s bivariate exponential and Marshall–Olkin bivariate exponential are used to model dependency between components. The results show that the reliability of the system with Freund’s bivariate exponential distribution has lower reliability. The circumstances that lead to them, namely load-sharing in the case of Freund, results in lower reliability. Finally, a numerical example is solved to evaluate the proposed model and sensitivity analysis is performed on the system reliability function. The obtained results show that because the proposed model is influenced by the dependency, compared to traditional models, it has the characteristic of leading to reduced time to (first) failure for achieving specified reliability.

A family of densities derived from the three-parameter Dirichlet process

2002 ◽  
Vol 39 (4) ◽  
pp. 764-774 ◽  
Author(s):  
Matthew A. Carlton
Keyword(s):  
State Space ◽  
Density Function ◽  
Dirichlet Process ◽  
Dirichlet Distribution ◽  
The State ◽  
Measurable Partition ◽  
Multivariate Distributions ◽  
The Family ◽  
Multivariate Density ◽  
Special Case

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

Correlated reliability and an application: Propulsive landing on Mars

2019 ◽  
Vol 233 (5) ◽  
pp. 826-846
Author(s):  
Hüseyin Sarper
Keyword(s):  
System Reliability ◽  
Simulation Method ◽  
The Other ◽  
Simulation Methods ◽  
Exponential Distributions ◽  
Numerical Examples ◽  
Beneficial Effects ◽  
Bivariate Exponential ◽  
System Life ◽  
Near Future

This article discusses reliability of landers and provides a review and examples of correlated reliability. Examples are cited to show generally beneficial effects of correlation in system reliability. Then, reliabilities of two near future landing systems are studied using two analytical (Downton, and Marshall & Olkin) bivariate exponential distributions and two simulation methods that incorporate correlation in reliability calculations. Both landing systems are composed of correlated two-unit subsystems. Numerical examples show mean system life, standard deviation of the system life, mean system life confidence interval, and reliability for each lander’s propulsive descent. Both simulation method results are in between the results obtained from the two analytical methods and Downton’s method yields the most conservative reliability. This article also shows how the Downton method–based reliability value can be predicted as a function of the reliabilities obtained from the other three methods. An up-to-date literature review of all related topics is also provided.

scholarly journals Maximal Percentages in Pólya's Urn

2015 ◽  
Vol 52 (1) ◽  
pp. 180-190 ◽  
Author(s):  
Ernst Schulte-Geers ◽  
Wolfgang Stadje
Keyword(s):  
Positive Probability ◽  
Urn Model ◽  
Pólya’S Urn

We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability, and derive several exact distributional results for the all-time maximal percentage.

Existence and positivity of the limit in processes with a branching structure

1999 ◽  
Vol 36 (1) ◽  
pp. 132-138
Author(s):  
M. P. Quine ◽  
W. Szczotka
Keyword(s):  
Stochastic Process ◽  
Branching Process ◽  
Random Variables ◽  
Random Variable ◽  
Natural Generalization ◽  
Partial Sums ◽  
Urn Model ◽  
Special Case ◽  
Independent And Identically Distributed

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

Sign in / Sign up

Export Citation Format

Share Document