Archimedean Copula Models Research Articles

Quantile regression based on the counting process approach under dependent truncated data

This paper focuses on analysing dependent truncated data. To address this, we draw inspiration from some literatures and employ the Archimedean copula model to establish a correlation between the truncation time and the survival time. Building on this, we refer to the U-statistics and the copula-graphic method to estimate the association parameter and the survival function. Subsequently, we proceed to build a quantile regression model and develop an estimation procedure for the quantile regression parameters using the counting process method. To further validate the efficacy of our approach, we conduct simulation experiments with specific settings of quantiles and correlations. In conclusion, we apply our method to analyse two datasets: one pertaining to HIV transfusion infections and the other related to retirement centres.

VALUE AT RISK ESTIMATION FOR STOCK PORTFOLIO USING THE ARCHIMEDEAN COPULA APPROACH

Investment is one of the many ways to achieve future profits. One form of investment that is widely made is stocks. The return obtained in investing in stocks is potentially higher than other investment alternatives, but the risks borne are amplified, so it is necessary to analyze these risks that may occur. In this study, the Archimedean copula method is used to estimate the Value at Risk on shares of PT Bank Rakyat Indonesia Tbk (BBRI) and PT Telekomunikasi Indonesia Tbk (TLKM) for the period September 1, 2021, to August 31, 2023. The stock data is used to determine the Archimedean copula model and calculate the estimated value of Value at Risk (VaR) on the stock return portfolio using the Archimedean copula approach. The Archimedean copula models used are the Clayton copula model, Gumbel copula, and Frank copula. Of the three Archimedean copula models, the best model was selected by looking at the largest Maximum Likelihood Estimation (MLE) value. In this study, the log-likelihood value of Clayton copula is 7.958, Gumbel copula is 6.663, and Frank copula is 8.398. Therefore, Frank copula is the best Archimedean copula model with the largest log-likelihood value of 8.398 for the said data. Then the VaR estimation is done with the Frank copula model. The Value at Risk estimation results based on the Frank copula model show maximum loss rates of -0.0277 at the 90% confidence level, -0.0363 at the 95% confidence level, and -0.0516 at the 99% confidence level.

Modeling dependence structure between green sukuk spread in Malaysia and the uncertainty factors before and during the COVID-19 pandemic

Purpose This paper aims to analyze the dependence structure between the Green Sukuk Spread in Malaysia and uncertainty factors from January 1, 2017, to May 23, 2023, covering two main periods: the pre-COVID-19 and the COVID-19 periods. Design/methodology/approach This study contributes to the current literature by explicitly modeling nonlinear dependencies using the Regular vine copula approach to capture asymmetric characteristics of the tail dependence distribution. This study used the Archimedean copula models: Student’s-t, Gumbel, Gaussian, Clayton, Frank and Joe, which exhibit different tail dependence structures. Findings The empirical results suggest that Green Sukuk and various uncertainty variables have the strongest co-dependency before and during the COVID-19 crisis. Due to external uncertainties (COVID-19), the results reveal that global factors, such as the Infect-EMV-index and the higher financial stress index, significantly affect the spread of Green Sukuk. Interestingly, in times of COVID-19, its dependence on Green Sukuk and the news sentiment seems to be a symmetric tail dependence with a Student’s-t copula. This result is relevant for hedging strategies, as investors can enhance the performance of their portfolio during the COVID-19 crash period. Originality/value This study contributes to a better understanding of the dependency structure between Green Sukuk and uncertainty factors. It is relevant for market participants seeking to improve their risk management for Green Sukuk.

Quantile Regression Based on the Weighted Approach with Dependent Truncated Data

This paper discusses the estimation of parameters in the quantile regression model for dependent truncated data. To account for the dependence between the survival time and the truncated time, the Archimedean copula model is used to construct the association. The parameters of the Archimedean copula model are estimated using certain existing approaches. An inference procedure based on a weighted approach is proposed, where the weights are set according to the variables of interest in the quantile regression model. The finite sample performance of the proposed approach is examined through simulations, and the method is applied to analyze two real datasets: the transfusion-related AIDS dataset and the retirement community center dataset.

The analysis of semi‐competing risks data using Archimedean copula models

In this paper, we derive the copula‐graphic estimator (Zheng and Klein) for marginal survival functions using Archimedean copula models based on competing risks data subject to univariate right censoring and prove its uniform consistency and asymptotic properties. We then propose a novel parameter estimation method based on the semi‐competing risks data using Archimedean copula models. Based on our estimation strategy, we propose a new model selection procedure. We also describe an easy way to accommodate possible covariates in data analysis using our strategies. Simulation studies have shown that our parameter estimate outperforms the estimator proposed by Lakhal, Rivest and Abdous for the Hougaard model and the model selection procedure works quite well. We fit a leukemia dataset using our model and end our paper with some discussion.

Time to Kidneys Failure Modeling in the Patients at Adama Hospital Medical College: Application of Copula Model.

Kidney failure is a common public health problem around the world. The vast majority of kidney failure cases in Sub-Saharan African nations, including Ethiopia, go undetected and untreated, resulting in practically certain mortality cases. This study was aimed primarily to model the time to (right and left) kidneys failure in the patients at Adama Hospital Medical College using the copula model. A retrospective cohort study. The copula model was used to examine join time to the right and left kidneys failure in the patients by specifying the dependence between the failure times. We employed Weibull, Gompertz, and Log-logistic marginal baseline distributions with Clayton, Gumbel, and Joe Archimedean copula families. This research comprised a total of 431 patients, out of which, 170 (39.4%) of the total patients failed at least one kidney during the follow-up period. Factors such as sex, age, family history of kidney disease, diabetes mellitus, hypertension, and obesity were found to be the most predictive variables for kidney failure in the patients. There was a 41 percent correlation between the patients' time to the right and left kidneys failure. The patients' kidney failure risk factors included being a male, older adult, obese, hypertensive, diabetic and also having a family history of kidney disease. The dependence between the patient's time to the right and left kidneys failure was strong. The best statistical model for describing the kidney failure datasets was the log-logistic-Clayton Archimedean copula model.

Bivariate Survival Copula Analysis of Glaucoma Patients during Blindness: Glaucoma Cases at Alert Hospital in Addis Ababa City of Ethiopia

Glaucoma is a worldwide problem that causes vision loss and even blindness, with a prevalence rate ranging from 1.9% to 15%. In Ethiopia, glaucoma is the fifth cause of blindness. This study aimed to explore the dependence between blindness of the right and the left eyes of glaucoma patients and assess the effects of the covariates under the dependence structure. A retrospective cohort study. The study population included the glaucoma patients at Alert hospital from January 1, 2018, to December 30, 2021. The copula model was used to estimate the time to the blindness of the right and the left eyes of the glaucoma patients by specifying the dependence between the event times. Out of 537 glaucoma patients, 224 (41.71%) became blind at least in one eye during the follow-up period. The results of the Clayton copula model revealed that factors, such as age, residence, diabetes mellitus, stage of glaucoma, and hypertension are considered the most prognostic factors for blindness in glaucoma patients. The findings also revealed that there was a strong dependence between the time to the blindness of the right and the left eyes in the glaucoma patients (τ = 0.43). Based on the obtained results, high age, urban residence, hypertension, diabetes mellitus, and higher stage of glaucoma were factors associated with time to the blindness in the glaucoma patients. There was also a dependence between the right and the left eyes of the glaucoma patients. The results revealed that the Clayton Archimedean copula model was the best statistical model for accurate description of glaucoma patients' datasets.

Seismic fragility models of a bridge system based on copula method

Seismic fragility assessment widely uses a probabilistic measure for assessing the seismic performance of structural components or systems. This article proposes a copula-based seismic fragility (CBSF) method to derive the system-level damage probabilities of reinforced concrete bridges and to consider the correlation among seismic demands of components. First, the marginal distribution functions of the random variables are calibrated, and three Archimedean copula models are considered. Subsequently, the relevant parameters of the copula models are estimated using the semi-parametric maximum likelihood method. Next, the damage probabilities of a bridge system are calculated based on the joint distribution model with the most suitable copula model and the calibrated marginal distribution functions. Finally, the CBSF method is used to estimate the damage probability of a simply supported box girder bridge. The performance of the CBSF method is validated by a comparison of fragility curves obtained using the CBSF method and the probabilistic seismic demand analysis (PSDA) method. The comparative results demonstrate that the fragility curves obtained by the CBSF method are better than those obtained using the PSDA method. The proposed CBSF model can serve as a tool for assessing the seismic performance of structures and estimating the economic loss of existing bridge systems.

In-home and out-of-home energy consumption by elderly and non-elderly households: a comparative analysis based on a copula model

This paper compares the energy consumption of elderly and non-elderly households, recognizing the ever-increasing importance of the aging population for energy policy. Decisions of the domestic (in-home) and transport (out-of-home) sectors at the household level are jointly incorporated by building a multivariate Hierarchical Archimedean Copula model with intra-sector and inter-sector dependencies. Data from a life-oriented behavioral survey in Japan are used, which includes 532 elderly households and 340 non-elderly households. A variety of life-related factors, together with built environment factors, are introduced into the above model. Modeling analysis results show that inter- and intra-sector dependencies are heterogeneous across households in terms of these factors. It is found that almost equal numbers of influential factors are sensitive and insensitive to age-based sample segmentation. Various similarities and dissimilarities across energy types between elderly and non-elderly households are revealed. The paper concludes with a discussion of the policy implications of the results.

Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

Outer power (OP) transformations of Archimedean generators are suggested to increase the modeling flexibility and statistical fitting capabilities of classical Archimedean copulas restricted to a single parameter. For OP-transformed Archimedean copulas, a formula for computing tail dependence coefficients is obtained, as well as two feasible OP Archimedean copula estimators are proposed and their properties studied by simulation. For hierarchical extensions of OP-transformed Archimedean copulas under the sufficient nesting condition, a new construction principle, efficient sampling and parameter estimation for models based on a single one-parameter Archimedean family are addressed. Special attention is paid to the case where the sufficient nesting condition simplifies to two types of restrictions on the corresponding parameters. By simulation, the convergence rate and standard errors of the proposed estimator are studied. Excellent tail fitting capabilities of OP-transformed hierarchical Archimedean copula models are demonstrated in a risk management application. The results show that the OP transformation is able to improve the statistical fit of exchangeable Archimedean copulas, particularly of those that cannot capture upper tail dependence or strong concordance, as well as the statistical fit of hierarchical Archimedean copulas, especially in terms of tail dependence and higher dimensions. Given how comparably simple it is to include OP transformations into existing exchangeable and hierarchical Archimedean copula models, OP transformations provide an attractive trade-off between computational effort and statistical improvement.

Estimation of the cumulative baseline hazard function for dependently right-censored failure time data

In this paper, we study the properties of a special class of frailty models when the frailty is common to several failure times. The models are closely linked to Archimedean copula models. We establish a useful formula for cumulative baseline hazard functions and develop a new estimator for cumulative baseline hazard functions in bivariate frailty regression models. Based on our proposed estimator, we present a graphical model checking procedure. We fit a leukemia data set using our model and end our paper with some discussions.

On bivariate Kumaraswamy-distorted copulas

We propose families of bivariate copulas based on the Kumaraswamy distortion of existing copulas. With the additional two parameters in the Kumaraswamy distribution, the induced copulas permit more flexibility in tail behaviors. The framework employed in this paper also provides a method for generating new Archimedean copula models. Two theorems linking the original tail dependence behaviors and those of the distorted copula are derived for distortions that are asymptotically proportional to the power transformation in the lower tail and to the dual-power transformation in the upper tail. We also derive explicit formulas for the Kendall’s τ coefficients, tail order parameters and tail order functions for the induced copulas when Gumbel, Clayton, Frank and Galambos are distorted. An empirical application is also presented.

Copula Model Selection for Vehicle Component Failures Based on Warranty Claims

In the automotive industry, it is important to know whether the failure of some car parts may be related to the failure of others. This project studies warranty claims for five engine components obtained from a major car manufacturer with the purpose of modeling the joint distributions of the failure of two parts. The one-dimensional distributions of components are combined to construct a bivariate copula model for the joint distribution that makes it possible to estimate the probabilities of two components failing before a given time. Ultimately, the influence of the failure of one part on the operation of another related part can be described, predicted, and addressed. The performance of several families of one-parameter Archimedean copula models (Clayton, Gumbel–Hougaard, survival copulas) is analyzed, and Bayesian model selection is performed. Both right censoring and conditional approaches are considered with the emphasis on conditioning to the warranty period.

Copula models for one-shot device testing data with correlated failure modes

Copula models have become one of the most popular tools, especially in finance and insurance, for modeling multivariate distributions in the past few decades, and they have recently received increasing attention for data analysis in reliability engineering and survival analysis. This paper considers two Archimedean copula models — the Gumbel-Hougaard copula and Frank copula — for analyzing one-shot device data with two correlated failure modes, which are collected from constant-stress accelerated life tests. A one-shot device is a unit that cannot be used again after a test, e.g., munitions, rockets, and automobile airbags. Only either left- or right-censored data are collected instead of the actual lifetimes of the devices under test. With the aid of Kendall’s tau correlation coefficient, initial values of the dependence parameter for the copula models are presented to determine maximum likelihood estimates of model parameters through a numerical approach. Furthermore, the proposed model can be used to examine whether the correlation between times to failure modes changes over stress levels. Real data from a survival experiment are also re-analyzed to illustrate the proposed methods.

Modelling the statistical dependence of rainfall variables using copula: A case study of Terengganu state

Extreme rainfall can have severe negative impacts possibly resulting in loss of life and property, hence the need for means of predicting the occurrence of extreme rainfall. One method that addresses extreme phenomena is the extreme value theory. In bivariate data, extreme value analysis can be approximated by the distribution of maximum likelihood estimation. This paper discusses the role of the copula in modeling the structure of dependencies between two rainfall variables, namely duration, and severity. Goodness-of-fit approaches are used to resolve whether or not to reject a parametric copula which then determines the best copula to utilise for the data set. In this paper the Akaike information criterion (AIC) is investigated for its ability to choose an appropriate copula model from Archimedean copula models. The analysis shows that the Frank copula is the best model to explain the dependency structure of the two variables discussed.

A Comparison of Archimedean Copula Models for approximating Bivariate Skew-Normal Distribution

This paper compares the performance of some Archimedean Copulas in approximating the bivariate skew-normal distribution. Our study shows Frank Copula is a better Archimedean Copula for approximating the bivariate skew-normal distribution.

Elliptical and Archimedean copula models: an application to the price estimation of portfolio credit derivatives

This paper explores the impact of elliptical and Archimedean copula models on the valuation of basket default swaps. We employ Monte Carlo simulation, in connection with the copula models, to estimate the default times and to calculate the swap payment legs and the cumulative swap premium. The numerical experiments reveal some sensitivity analysis on the impact of swap parameters on the fair prices of the 𝑛th-to-default swaps. Finally, using the results presented, an appropriate choice of copula model can be made based on the computation time of the valuation process, and such a choice hugely affects the quantitative risk analysis of the portfolio.

Conditional tail independence in Archimedean copula models

AbstractConsider a random vector $\textbf{U}$ whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $\textbf{U}$ , conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.

Analysis of clustered failure time data with cure fraction using copula.

Clustered survival data in the presence of cure has received increasing attention. In this paper, we consider a semiparametric mixture cure model which incorporates a logistic regression model for the cure fraction and a semiparametric regression model for the failure time. We utilize Archimedean copula (AC) models to assess the strength of association for both susceptibility and failure times between susceptible individuals in the same cluster. Instead of using the full likelihood approach, we consider a composite likelihood function and a two-stage estimation procedure for both marginal and association parameters. A Jackknife procedure that takes out one cluster at a time is proposed for the variance estimation of the estimators. Akaike information criterion is applied to select the best model among ACs. Simulation studies are performed to validate our estimating procedures, and two real data sets are analyzed to demonstrate the practical use of our proposed method.

Modeling cause-of-death mortality using hierarchical Archimedean copula

ABSTRACTStudying changes in cause-specific (or competing risks) mortality rates may provide significant insights for the insurance business as well as the pension systems, as they provide more information than the aggregate mortality data. However, the forecasting of cause-specific mortality rates requires new tools to capture the dependence among the competing causes. This paper introduces a class of hierarchical Archimedean copula (HAC) models for cause-specific mortality data. The approach extends the standard Archimedean copula models by allowing for asymmetric dependence among competing risks, while preserving closed-form expressions for mortality forecasts. Moreover, the HAC model allows for a convenient analysis of the impact of hypothetical reduction, or elimination, of mortality of one or more causes on the life expectancy. Using US cohort mortality data, we analyze the historical mortality patterns of different causes of death, provide an explanation for the ‘failure’ of the War on Cancer, and evaluate the impact on life expectancy of hypothetical scenarios where cancer mortality is reduced or eliminated. We find that accounting for longevity improvement across cohorts can alter the results found in existing studies that are focused on one single cohort.