The effect of dependence on distribution of the functions of random variables

Dolati, Ali; Roozegar, Rasool; Ahmadi, Najmeh; Shishebor, Z.

doi:10.1080/03610926.2016.1242740

Cited by 8 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If the continuous stress and strength r.v.s X ⇠ F and Y ⇠ G are linked by a copula C, then the stress-strength parameter R is given by (see [10])…”

Section: Reliability For Stress-strength Model With Dependent Vari-ablesmentioning

confidence: 99%

“…If additionally X and Y are both positive r.v.s, then the stress-strength parameter R is given also by (see [10] again)…”

Section: Reliability For Stress-strength Model With Dependent Vari-ablesmentioning

confidence: 99%

“…Exploiting formulas (5) or (6), in [10] the expressions of the reliability parameter have been explicitly or numerically found for some possible choices of F and G (namely, uniform and exponential) and the linking copulas (Farlie-Gumbel-Morgenstern, Frank, Clayton and Gumbel).…”

Section: Reliability For Stress-strength Model With Dependent Vari-ablesmentioning

confidence: 99%

“…In [9] the problem of estimation of the reliability parameter is considered when the Farlie-Gumbel-Morgenstern copula is used to link stress and strength variables, whose marginal distributions both belong to the Burr system. More recently, in [10], an ampler study on the effect of statistical dependence on the distribution of the functions of r.v.s deals also with the computation of R when several statistical distributions are chosen for both X and Y and for various copula structures.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

Barbiero¹

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

View full text Add to dashboard Cite

Abstract. The stress-strength model is a basic modeling tool in reliability analysis. In simple terms, it considers a component (or a system) with an intrinsic strength Y , which is subjected to a stress X; the component works if and only if Y is greater than X. If stress and strength are regarded as random variables, then the probability that the component works is given by P (X < Y ) and is usually called "reliability parameter". Statistical independence is usually assumed between the two random variables X and Y : for this case, the literature on this topic is particularly rich. This strong assumption, in fact, makes the calculation and estimation of the reliability parameter R more tractable. However, this hypothesis is not always verified in practice, and this translates into an over-or under-estimation of R. To avoid this drawback, statistical dependence can be introduced and modeled between X and Y , for example resorting to copulas. In some recent works, the problem of computing and estimating R is considered when the stress and strength variables, belonging to the same parametric family of distributions, are linked by a specific copula. In this work, we further consider the computational issues related to this copula approach applied to the stress-strength model, when other families of copulas are selected. A sort of sensitivity analysis is performed in order to assess how the value of the reliability parameter is affected by the choice of the copula binding X and Y together and of its parameters. As a limit case, we consider the situation where no information on the dependence structure of the stress and strength margins is available and try to provide lower and upper bounds for R.

show abstract

“…If the continuous stress and strength r.v.s X ⇠ F and Y ⇠ G are linked by a copula C, then the stress-strength parameter R is given by (see [10])…”

Section: Reliability For Stress-strength Model With Dependent Vari-ablesmentioning

confidence: 99%

“…If additionally X and Y are both positive r.v.s, then the stress-strength parameter R is given also by (see [10] again)…”

Section: Reliability For Stress-strength Model With Dependent Vari-ablesmentioning

confidence: 99%

Section: Reliability For Stress-strength Model With Dependent Vari-ablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

Barbiero¹

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

View full text Add to dashboard Cite

show abstract

“…To relax independence assumption, it is necessary to develop a framework for modeling dependence structures of random vectors in more general sense. To do so, Dolati et al (2017) develop the distribution for X/Y in which both X and Y are positive.…”

Section: Introductionmentioning

confidence: 99%

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Pho

et al. 2019

JRFM

View full text Add to dashboard Cite

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

show abstract

A copula-based exponential probabilistic model for factor-dependence social sustainability assessment

Khosravi,

Izbirak

2023

Environ Dev Sustain

View full text Add to dashboard Cite

The effect of dependence on distribution of the functions of random variables

Cited by 8 publications

References 12 publications

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

A copula-based exponential probabilistic model for factor-dependence social sustainability assessment

Contact Info

Product

Resources

About