Copula Representations for the Sum of Dependent Risks: Models and Comparisons

Navarro, Jorge; Sarabia, José Marı́a

doi:10.1017/s0269964820000649

Cited by 8 publications

(6 citation statements)

References 44 publications

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“…This kind of representations are called distorted distributions , and they are studied in the following subsection. A similar representation can be obtained when just G is an exponential distribution (see theorem 2.2 in Navarro and Sarabia 8 ). In the bivariate case such a representation can be stated as follows.…”

Section: Preliminariesmentioning

confidence: 55%

“…

<

) represents a positive (negative) dependence (see, e.g., Nelsen 15 ). If we also assume that X and Y have a common exponential distribution with hazard rate

λ > 0

, then a straightforward calculation from (3) gives

\overline{H} (t) = {\overline{q}}_{θ} (\overline{F} (t))

with

\overline{F} (t) = e^{- λ t}

for

t \geq 0

and

{\overline{q}}_{θ} (u) = u - u \log u + θ u [- 3 + 3 u - \log u - 2 u \log u]

for

u \in [0, 1]

(see also Navarro and Sarabia 8 ). As expected, in the independent case (

θ = 0

), we obtain

{\overline{q}}_{0} (u) = u - u \log u

for

u \in [0, 1]

.…”

Section: Preservation Results For the Ifr Classmentioning

confidence: 99%

“…Then, from Cherubini et al 13 (see also Navarro and Sarabia 8 ), the distribution function H of the sum

S = X + Y

can be obtained as

H (t) = \int_{- \infty}^{\infty} f (x) \Pr (Y \leq t - x | X = x) d x = \int_{- \infty}^{\infty} f (x) \partial_{1} C (F (x), G (t - x)) d x,

where

\partial_{1} C

represents the partial derivative of C with respect to its first argument. If X and Y are nonnegative, then the limits in the integral reduces to 0 and t , respectively.…”

Section: Preliminariesmentioning

confidence: 99%

“…The purpose of the present paper is to investigate this insight in more detail, in order to provide new conditions for preservation of some of the most well known aging classes under sums of dependent random variables. The new statements can be added to similar ones recently obtained in Navarro and Sarabia 8 for the IFR/DFR classes. The results described here are mainly based on preservation properties for distorted distributions and C‐convolutions, which are copula representations for the distributions of sums of dependent random variables (see Section 2).…”

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations

Preservation of ILR and IFR aging classes in sums of dependent random variables

Navarro

Pellerey

2021

Appl Stoch Models Bus & Ind

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Closure of aging classes with respect to sums is a relevant topic in different areas of applied probability. In the case of independent random variables (i.e., for convolutions) this preservation property has been proved in the literature for a number of classes such as the increasing in likelihood ratio (ILR) and the increasing failure rate (IFR) classes. These results were applied, for example, to sums of life lengths in reliability theory and to sums of incomes/returns in economic/risk studies. However, in many practical situations the independence assumption is unrealistic. In the present paper, we provide conditions such that these closure properties are satisfied by the ILR and IFR classes when the assumption of independence is dropped. The classical copula approach is used to model the dependence structure, but other dependence models, such as relevation transforms and load sharing models, are considered as well. Several illustrative examples and counterexamples show how to use the presented theoretical results and which preservation results do not hold.

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Section: Preliminariesmentioning

confidence: 55%

“…

<

) represents a positive (negative) dependence (see, e.g., Nelsen 15 ). If we also assume that X and Y have a common exponential distribution with hazard rate

λ > 0

, then a straightforward calculation from (3) gives

\overline{H} (t) = {\overline{q}}_{θ} (\overline{F} (t))

with

\overline{F} (t) = e^{- λ t}

for

t \geq 0

and

{\overline{q}}_{θ} (u) = u - u \log u + θ u [- 3 + 3 u - \log u - 2 u \log u]

for

u \in [0, 1]

(see also Navarro and Sarabia 8 ). As expected, in the independent case (

θ = 0

), we obtain

{\overline{q}}_{0} (u) = u - u \log u

for

u \in [0, 1]

.…”

Section: Preservation Results For the Ifr Classmentioning

confidence: 99%

“…Then, from Cherubini et al 13 (see also Navarro and Sarabia 8 ), the distribution function H of the sum

S = X + Y

can be obtained as

H (t) = \int_{- \infty}^{\infty} f (x) \Pr (Y \leq t - x | X = x) d x = \int_{- \infty}^{\infty} f (x) \partial_{1} C (F (x), G (t - x)) d x,

where

\partial_{1} C

represents the partial derivative of C with respect to its first argument. If X and Y are nonnegative, then the limits in the integral reduces to 0 and t , respectively.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Preservation of ILR and IFR aging classes in sums of dependent random variables

Navarro

Pellerey

2021

Appl Stoch Models Bus & Ind

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show abstract

“…where F1 and F2 are the marginal survival functions of X 1 and X 2 , respectively, f 1 is the density function of X 1 (assuming its existence) and C is the survival copula of the vector X. This expression, obtained in Cherubini et al (2011), is a key tool in our results (see, also Cherubini et al (2016) and Navarro and Sarabia (2020), for additional examples of C-convolutions).…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

On sums of dependent random lifetimes under the time-transformed exponential model

Navarro

Pellerey

Mulero

2022

TEST

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For a given pair of random lifetimes whose dependence is described by a time-transformed exponential model, we provide analytical expressions for the distribution of their sum. These expressions are obtained by using a representation of the joint distribution in terms of bivariate distortions, which is an alternative approach to the classical copula representation. Since this approach allows one to obtain conditional distributions and their inverses in simple form, then it is also shown how it can be used to predict the value of the sum from the value of one of the variables (or vice versa) by using quantile regression techniques.

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Simulations and predictions of future values in the time-homogeneous load-sharing model

Buono

Navarro

2023

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Copula Representations for the Sum of Dependent Risks: Models and Comparisons

Cited by 8 publications

References 44 publications

Preservation of ILR and IFR aging classes in sums of dependent random variables

Preservation of ILR and IFR aging classes in sums of dependent random variables

On sums of dependent random lifetimes under the time-transformed exponential model

Simulations and predictions of future values in the time-homogeneous load-sharing model

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