COPULA REPRESENTATIONS FOR THE SUM OF DEPENDENT RISKS: MODELS AND COMPARISONS

2020 ◽  
pp. 1-21
Author(s):  
Jorge Navarro ◽  
José María Sarabia
Keyword(s):  
Hazard Rate ◽  
Marginal Distribution ◽  
Rate Function ◽  
Distribution Functions ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Limiting Behavior ◽  
Dependent Risks ◽  
Analytical Expressions ◽  
Theoretical Results

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (4) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (04) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X 1,…,X n be independent exponential random variables with X i having hazard rate . Let Y 1,…,Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i=1 n λ i )1/n , the geometric mean of the λis. Let X n:n = max{X 1,…,X n }. It is shown that X n:n is greater than Y n:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n:n and an upper bound on the hazard rate function of X n:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference 65, 203–211), which are in terms of the arithmetic mean of the λ i s. Furthermore, let X 1 *,…,X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λ i ∗, i = 1,…,n. It is proved that if (logλ1,…,logλ n ) weakly majorizes (logλ1 ∗,…,logλ n ∗, then X n:n is stochastically greater than X n:n ∗.

Reliability Concepts

2018 ◽  
pp. 1-22
Keyword(s):  
Hazard Rate ◽  
Failure Process ◽  
Reliability Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Hazard Rate Function ◽  
Reliability Engineering ◽  
Probability Models ◽  
Engineering Design Process ◽  
Basic Concepts

The first chapter introduces basic concepts of Reliability and their relationships. Four probability functions—reliability function, cumulative distribution function, probability density function, and hazard rate function—that completely characterize the failure process are defined. Three failure rates—MTBF, MTTF, MTTR—that play important role in reliability engineering design process are explained here. The three patterns of failures, DFR, CFR, and IFR, are discussed with reference to the bathtub curve. Two probability models, Exponential and Weibull, are presented. Series and parallel systems and application areas of reliability are also presented.

Bounds for the hazard rate and the reversed hazard rate of the convolution of dependent random lifetimes

2019 ◽  
Vol 56 (4) ◽  
pp. 1033-1043 ◽  
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Hazard Rate ◽  
Rate Function ◽  
Semiparametric Models ◽  
Hazard Rate Function ◽  
Reversed Hazard Rate ◽  
Parametric And Semiparametric Models ◽  
Reversed Hazard Rate Function

AbstractAn upper bound for the hazard rate function of a convolution of not necessarily independent random lifetimes is provided, which generalizes a recent result established for independent random lifetimes. Similar results are considered for the reversed hazard rate function. Applications to parametric and semiparametric models are also given.

A stochastic process for modeling failures of a system having a non-monotonic hazard rate function

2019 ◽  
Vol 233 (5) ◽  
pp. 731-746
Author(s):  
Lucianne Varn ◽  
Stefanka Chukova ◽  
Richard Arnold
Keyword(s):  
Stochastic Process ◽  
Hazard Rate ◽  
Repair Process ◽  
Rate Function ◽  
Hazard Rates ◽  
Failure Rates ◽  
Hazard Rate Function ◽  
Repairable Systems ◽  
System Failures ◽  
Quantities Of Interest

Reliability literature on modeling failures of repairable systems mostly deals with systems having monotonically increasing hazard/failure rates. When the hazard rate of a system is non-monotonic, models developed for monotonically increasing failure rates cannot be effectively applied without making assumptions on the types of repair performed following system failures. For instance, for systems having bathtub-shaped hazard rates, it is assumed that during the initial, decreasing hazard rate phase, all repairs are minimal. These assumptions on the type of general repair can be restrictive. In order to relax these assumptions, it has been suggested that general repairs in the initially decreasing phase can be modeled as “aging” the system. This approach however does not preserve the order of effectiveness of the types of general repair as defined in the literature. In this article, we develop a set of models to address these shortcomings. We propose a new stochastic process to model consecutive failures of repairable systems having non-monotonic, specifically bathtub-shaped, hazard rates, where the types of general repair are not restricted and the order of the effectiveness of the types of repair is preserved. The proposed models guarantee that a repaired system is at least as reliable as one that has not failed (or equivalently one that has been minimally repaired). To illustrate the models, we present multiple examples and simulate the failure-repair process and estimate the quantities of interest.

scholarly journals Slashed Lomax Distribution and Regression Model

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1877
Author(s):  
Huihui Li ◽  
Weizhong Tian
Keyword(s):  
Regression Model ◽  
Hazard Rate ◽  
Rate Function ◽  
Model Parameters ◽  
Asymmetric Distribution ◽  
Hazard Rate Function ◽  
Simulation Studies ◽  
Ecm Algorithm ◽  
Lomax Distribution ◽  
Skewness And Kurtosis

In this article, the slashed Lomax distribution is introduced, which is an asymmetric distribution and can be used for fitting thick-tailed datasets. Various properties are explored, such as the density function, hazard rate function, Renyi entropy, r-th moments, and the coefficients of the skewness and kurtosis. Some useful characterizations of this distribution are obtained. Furthermore, we study a slashed Lomax regression model and the expectation conditional maximization (ECM) algorithm to estimate the model parameters. Simulation studies are conducted to evaluate the performances of the proposed method. Finally, two sets of data are applied to verify the importance of the slashed Lomax distribution.

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