The Mean of Marshall–Olkin-Dependent Exponential Random Variables

2015 ◽  
pp. 33-50 ◽  
Author(s):  
Lexuri Fernández ◽  
Jan-Frederik Mai ◽  
Matthias Scherer
Keyword(s):  
Random Variables ◽  
Exponential Random Variables ◽  
The Mean

ORDERING CONVOLUTIONS OF HETEROGENEOUS EXPONENTIAL AND GEOMETRIC DISTRIBUTIONS REVISITED

2010 ◽  
Vol 24 (3) ◽  
pp. 329-348 ◽  
Author(s):  
Tiantian Mao ◽  
Taizhong Hu ◽  
Peng Zhao
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Mean Residual Life ◽  
Reversed Hazard Rate ◽  
Exponential Random Variables ◽  
The Mean ◽  
Geometric Variables ◽  
Fail Safe

Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and $(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ under which Sn(a1, …, an) and $S_{n}(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.

Improving the reliability of confidence limits for the mean of positive random variables

SIMULATION ◽  
1990 ◽  
Vol 55 (3) ◽  
pp. 175-177
Author(s):  
Do Le Minh ◽  
Y.S. Sherif
Keyword(s):  
Random Variables ◽  
Confidence Limits ◽  
The Mean

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (4) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

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.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (04) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

scholarly journals A Scalar Expected Value of Intuitionistic Fuzzy Random Individuals and Its Application to Risk Evaluation in Insurance Companies

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Chunquan Li ◽  
Jianhua Jin
Keyword(s):  
Risk Evaluation ◽  
Random Variables ◽  
Expected Value ◽  
Insurance Companies ◽  
Claim Amount ◽  
Fuzzy Random Variables ◽  
Intuitionistic Fuzzy ◽  
The Mean ◽  
Fuzzy Random ◽  
Mean Chance

Randomness and uncertainty always coexist in complex systems such as decision-making and risk evaluation systems in the real world. Intuitionistic fuzzy random variables, as a natural extension of fuzzy and random variables, may be a useful tool to characterize some high-uncertainty phenomena. This paper presents a scalar expected value operator of intuitionistic fuzzy random variables and then discusses some properties concerning the measurability of intuitionistic fuzzy random variables. In addition, a risk model based on intuitionistic fuzzy random individual claim amount in insurance companies is established, in which the claim number process is regarded as a Poisson process. The mean chance of the ultimate ruin is investigated in detail. In particular, the expressions of the mean chance of the ultimate ruin are presented in the cases of zero initial surplus and arbitrary initial surplus, respectively, if individual claim amount is an exponentially distributed intuitionistic fuzzy random variable. Finally, two illustrated examples are provided.

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