Clayton Copula Research Articles

Introducing Copula Functions to Estimate the Reliability of Dependent Mechanical Systems

This paper addresses the challenge of assessing the reliability of complex mechanical systems where components are inherently correlated in their failure modes. Traditionally, the assumption of independence among these components has been employed, but it often fails to capture the real-world complexities. To overcome this limitation, copula functions are introduced as a robust methodology for modeling the dependent relationships between correlated variables within mechanical systems. This paper aims to demonstrate the utility of copulas in estimating system reliability while accounting for these dependencies. The results reveal that the Clayton copula emerges as the most suitable model for representing dependence in such systems. Importantly, the reliability estimates obtained through copula-based methods not only reflect the complex interdependencies accurately but also align with the principles of the boundary theory of reliability. This research underscores the potential of copula-based reliability estimation as a valuable alternative, offering a more comprehensive and precise assessment of reliability in complex mechanical systems and holding significant promise for practical engineering applications. This framework allows the consideration of dependence among the observed variables that is usually overlooked in engineering practice.

PROSPECTIVE RESERVE AND FULL PRELIMINARY TERM RESERVE ON ENDOWMENT LAST SURVIVOR LIFE INSURANCE USING CLAYTON COPULA

Combined life insurance is a type of insurance that protects two or more people who are related by family and is divided into two, namely joint-life life insurance and last-survivor life insurance. The last survivor life insurance is a condition of life insurance that will continue if there is at least one of all insurance participants who is still alive and will stop if all insurance participants die. The insurance company has to pay the benefit to the heirs of the insurance participant. When a claim occurs, the insurance company must prepare the reserve fee. The purpose of this research is to determine the amount of premium reserve of endowment last-survivor life insurance using prospective reserve and full preliminary term reserve. Full preliminary term reserve is one of the modified premium reserve calculations from Zillmer Reserve. To determine prospective reserve and full preliminary term reserve using the initial life annuity, single premium, and annual premium. Whereas the initial life annuity is influenced by the combined life and death opportunity of the insurance participants. Furthermore, the combined life and death opportunity of insurance participants will be obtained from Clayton copula and to obtain the parameter of Clayton copula, Rstudio software is used. Based on the result, the value of prospective reserves and full preliminary term reserves has increased every year and prospective reserves produce a greater value than full preliminary term reserves. If the insurance company uses this reserve calculation, the reserve that the company must prepare will increase every year. This is useful for insurance companies in predicting the amount of reserves they must have.

Bayesian ridge estimators based on a vine copula-based prior in Poisson and negative binomial regression models

Poisson and negative binomial regressions are popular methods for modelling the relationship between a count variable and explanatory variables. In the presence of multicollinearity, ridge regression is an alternative to the maximum likelihood estimation for regression coefficients. Furthermore, ridge estimators are interpreted as the Bayesian posterior mean (or mode) when the regression coefficients follow a multivariate normal prior. However, using the multivariate normal prior may not effectively estimate regression coefficients, especially in the presence of interaction terms. This study proposes vine copula-based priors for Bayesian ridge estimators in Poisson and negative binomial regression models. The simulations and data analysis results indicate that for the Poisson model with equidispersion and the negative binomial model with overdispersion, the Clayton and Gumbel copula priors of the Archimedean family achieve superior performance than the multivariate normal prior and Gaussian copula prior.

Modelling Risk Dependencies in Insurance Using Survival Clayton Copula

Our aim in this paper is to show the use of survival Clayton copula as a suitable tool for modelling risk dependencies in insurance. A purpose-built simulation of an adequate upper tail dependence can be an important part of the aggregation of risks in an insurer’s internal models. The occurrence of extreme values of the aggregate random variable might have a very negative impact on the insurer when securing coverage of unexpected losses. The upper conditional quantile exceedance probability of the copula is a suitable indicator. In addition an analysis of its effect on the level of modelling of the risk scenario is available. This effect is measured using the Tail Value at Risk of the aggregate random variable. To simplify our description of the given principle for aggregating risks we will in this paper only consider the two-dimensional case. The programming language R was used to simulate the values of the joint distribution of the marginal random variables.

Towards the sustainable development of water security: A new copula-based risk assessment system

Water security is deeply intertwined with ecosystems, society and the economy, and is a frequent focal point of sustainable development. This study investigates the water security (WS) of Inner Mongolia, a dryland region in northern China. Specifically, an indicator system integrating the principles of Stability (S), Cooperation (C), and Resilience (R) was constructed to quantitatively and comprehensively assess water security. Risk probability was calculated by the constructed system by combining its output with a copula function. The findings revealed that: (1) WS-SCR levels in the study area gradually increased from 2001 to 2021. The periods multi-year average was 0.42, which is at the threshold level of security; (2) Based on a standard deviation ellipse, the spatial distribution of WS-SCR levels changed from northeast to southwest and back to northeast. The centroid of change shifted from east to west and back to east; (3) Key drivers influencing water security included GDP per capita (S3), agricultural irrigation water consumption per unit area (C4), comprehensive water use per capita (C5), reclaimed water use rate (C8), fertilizer application per unit of farmland (R4), and population density (R5); (4) The S-C, S-R, and C-R relationships within the study area were best modelled by the Clayton copula function, whereas the S-C-R relationship was most accurately modelled by using a Frank copula function. The three-dimensional joint risk probability S-C-R (S≤0.4, C≤0.4, R≤0.4) is 0.226; the probability that R was insecure increased with increasing S or C index values. This study provides a scientific foundation for the long-term planning and management of sustainable water security development in Eurasian drylands.

Unraveling The Dynamics Of Foreign Direct Investment And Economic Activity In Brazil: A Copula-Based Analysis

This study analyzes the relationship between FDI and the IBC-Br from 2003 to 2022. Our objective is to examine the effect of FDI on Brazil´s economic activities. Thus, we look also for a connection between the Brazilian Central Bank’s leadership in each period and the FDI effect on IBC-Br. To explore this, we employ an approach based on the Granger Causality Test and copula models that encompass Gaussian, t-Student, Gumbel, Frank, and Clayton functions. Our findings indicate that in the short term, FDI exerts some influence on IBC-Br, while the reverse relationship is not observed. Furthermore, using the Clayton Copula function, we identify the dependency of causal relationships between the variables, particularly in extreme economic scenarios. These discoveries contribute to a more comprehensive understanding of the dynamics between FDI and economic activity in the different periods of the Brazilian Central Bank’s leadership.

20 Evaluating intermediate endpoints for overall survival in metastatic renal cell carcinoma treated with immune checkpoint inhibitors: an IMDC study

Abstract Background In Phase 3 trials, the assessment for primary endpoint of overall survival (OS) necessitates extended follow-up periods, a larger pool of events, and higher associated costs. Hence, we sought to determine whether shorter intermediate (IEs) such as Objective Response (OR), Time to Treatment Failure (TTF) and Time to Next Therapy (TTNT) are associated with OS in patients receiving immune checkpoint (ICI)-based therapy. Methods We included all International mRCC Database Consortium (IMDC) patients who received contemporary first line(1L) ICI from 2013 to 2023. IEs were defined from ICI start until drug cessation or death for TTF, and initiation of next line or death for TTNT, or censored at date of last follow-up. OR included investigators-assessed complete or partial response per RECIST1.1 criteria. First, we assessed the endpoint correlations across all follow-up times of individual patient data using Kendall’s Tau (KT) correlation by a Clayton copula. A KT >0.49 indicates a strong correlation (Wicklin R., 2023). Then we evaluated associations of OS with TTF and TTNT status at the 6-month landmark using Cox regression adjusting for IMDC risk groups, metastatic sites, histology, age, and prior nephrectomy, stratified by ICI type and years of ICI start. The KT correlation was estimated by the R GJRM package (https://www.r-project.org/). All other statistical analyses were done with SAS 9.4 software (SAS Institute, Cary, NC). Results The cohort consisted of 1667 patients with a median follow-up of 15.4 months (IQR: 7.1-28.6). For the 6-month landmark analysis in OS, patients who died or had less than 6 months of follow-up were excluded, affecting 278 for OR, 373 for TTF, and 380 for TTNT. Median age at 1L start was 63 years (IQR: 56-70), with 73% being male and 65% undergoing nephrectomy before starting 1L. A total of 1132 patients received dual ICI, while 535 received an ICI+TKI combination. Over the entire follow-up of individual patient data, the Kendall’s Tau correlation was 0.49 (95%CI: 0.45-0.52) for TTF with OS and 0.67 (95%CI: 0.64-0.69) for TTNT with OS. The Kendall’s Tau correlations between endpoints were also similar in subgroup analyses by IMDC risk group and type of regimen and the strongest in the ICI+TKI subgroup with a Kendall’s Tau correlation of 0.73(0.66-0.78). On the 6-month OS landmark analysis (Table), patients who discontinued their 1L regimen had poor OS (adjusted HR: 2.77 (95%CI: 2.16-3.55)). Additionally, those who transitioned to a 2L therapy within the first 6 months showed worse OS, reflected by an HR of 2.82 (2.22-3.59). Patients who did not have an objective response also had worse OS (adjusted HR: 2.74 (95%CI: 2.15-3.49)). On the other hand, patients with an objective response had an 18-month OS rate (equivalent of 12-month OS post the 6-month landmark) of 91% (95%CI: 88%-93%) vs 74% (95%CI: 70%-78%) for those with no response at 6 months. Table. Landmark analysis of OS from 6 months of post therapy initiation, according to OR, TTF and TTNT event status at 6 months, in whole cohort and by IMDC and treatment groups. Conclusions Our study explored TTF, TTNT and OR as clinical IEs for OS. TTNT demonstrated the strongest association with OS, particularly in the ICI+TKI subgroup, making it a potentially clinically meaningful intermediate endpoint for evaluating treatment efficacy in ICI-based regimens.

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.

Bivariate Lomax Distribution Based on Clayton Copula: Estimation and Prediction

Lomax distribution has been studied by many statisticians due to its important role in reliability modeling and lifetime testing. The bivariate Lomax distribution is an important lifetime distribution in survival analysis. In this article, a bivariate Lomax distribution is constructed based on Clayton copula. The joint probability density function and the joint cumulative distribution function are derived in closed forms. Some properties of this bivariate distribution are discussed. The maximum likelihood and Bayes estimators for the unknown parameters are derived. Also, the maximum likelihood and Bayesian two-sample prediction for the future observations are obtained. The performance of the proposed bivariate distribution is examined using a simulation study. Finally, one real data set under the proposed distribution is considered to illustrate its flexibility and applicability for real-life applications.

BHF and copula models in small area estimation for household per capita expenditure in Bogor District

Small area statistics are required when the sample size is small to produce estimates with adequate precision. The assumptions underlying Battese Harter Fuller (BHF) unit-level models may often be unrealistic in some applications. Copula is an alternative approach when the assumptions are violated. This research discusses the performance of BHF and Copula in small area estimation (SAE) for estimating household per capita expenditure in sub-district levels. This study presents household per capita expenditure, which has a skewed distribution. Due to the fact that the data contains outliers, an appropriate method to handle outliers is also considered. In this research, the Gaussian and the Clayton Copulas are used. The results showed that the performance of BHF was better than Gaussian and Clayton Copulas, as indicated by small root mean square error (RMSE) with an average of 1.14, while the average RMSE of Gaussian copula was 2.71 and Clayton copula was 2.63. Furthermore, the coefficient of variation (CV) of BHF was also smaller compared to Gaussian and Clayton Copulas, and the resulting estimates can be categorized as reliable based on the CV of less than 25%.

Pricing CDS index tranches under thinning-dependence structure with regime switching

In this paper, we consider a portfolio credit risk model with the thinning-dependence structure under the case with regime switching and the one with no regime switching. We obtain the explicit expressions for the copulas of the default times. Based on the quotation convention of CDS index tranches, we derive some closed-form expressions for the pricing formulas for the CDS index tranche via the copulas and the marginal distribution functions of the default times. As an empirical example, we consider the pricing of the CDX NA IG Series 25. We demonstrate how the model parameter can be calibrated based on market data and how the numerical results can be calculated via the proposed pricing formulas. Compared to the empirical results under the pricing model with some commonly used copulas, such as the Clayton copula, the Gumbel copula, and the Frank copula, we find that the proposed pricing model with regime switching can provide a valuable reference for the market quotes for the CDX NA IG Series 25.

Advanced Copula-Based Models for Type II Censored Data: Applications in Industrial and Medical Settings

Copula models are increasingly recognized for their ability to capture complex dependencies among random variables. In this study, we introduce three innovative bivariate models utilizing copula functions: the XLindley (XL) distribution with Frank, Gumbel, and Clayton copulas. The results highlight the fundamental characteristics and effectiveness of these newly introduced bivariate models. Statistical inference for the distribution parameters is conducted using a Type II censored sampling design. This employs maximum likelihood and Bayesian estimation techniques. Asymptotic and credible confidence intervals are calculated, and numerical analysis is performed using the Markov Chain Monte Carlo method. The proposed methodology’s applicability is illustrated by analyzing several real-world datasets. The initial dataset examines burr formation occurrences and consists of two observation sets. Additionally, the second and third datasets contain medical information. The second dataset focuses on diabetic nephropathy, while the third dataset explores infection and recurrence time among kidney patients.

Bayesian mixed model for survival data with semicompeting risks based on the Clayton copula

Bayesian mixed model for survival data with semicompeting risks based on the Clayton copula

Estimate rainfall return period based on maximum pipe flow: A new framework to address mutual relationship

The unreasonable assumption that the rainfall return period is similar to the runoff return period may lead to unreasonable drainage system design. Hence, this study proposes a computational framework to calculate the rainfall return period according to the runoff return period. Based on the Akaike Information Criterion (AIC) and the optimal probability distribution functions, a novel approach was proposed to combine the rainfall and maximum pipe flow return periods. Copula functions were employed to construct a joint probability distribution for calculating the rainfall return period using the maximum pipe flow return period. Thus, this study reached the following results. (I) The Gamma, Logistic, and Weibull distributions provided better descriptions for the rainfall and maximum pipe flow probability distributions. The Gaussian, Gumbel, and Survival Clayton Copula functions were shown to be the optimal choices. (II) A calculation method for the rainfall return period was constructed based on the Satisfaction Percentage (SP). For instance, using a 2-year as the maximum pipe flow return period, the rainfall return period was determined to be 4.705 years (60 min of rainfall duration, Maluan community) with an SP value of 0.95. (III) The study area showed non-significant impacts (p < 0.05) in selecting the rainfall return periods. Therefore, simulations can be conducted in regions with missing pipe network data using similar underlying surface conditions from other study areas.

New Bivariate Copulas via Lomax Distribution Generated Distortions

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns.

Enhancing flood risk assessment in the Johor River Basin through trivariate copula

ABSTRACT Copulas are a vital statistical tool, particularly in hydrology, for understanding complex relationships among flood characteristics. This study focuses on three key flood features: peak discharge, flood volume, and flood duration, using trivariate copulas to capture their interdependencies. This is crucial because bivariate and univariate analyses fall short in considering all three factors simultaneously. To handle extreme flood values, L-moment is proposed over maximum likelihood estimation and inference function margin due to its enhanced reliability and susceptibility to outliers and extreme values. Akaike information criterion was employed to identify the best-fit marginal distribution and copula. The Lognormal distribution effectively models peak discharge, while Weibull and generalized extreme value distributions fit flood volume and duration best, respectively. Various copula families, including elliptical and Archimedean, are assessed, where Clayton copula emerge as the most suitable. This analysis demonstrates that when more flood features are considered together, the return period increases, indicating the reduced likelihood of occurrence. The trivariate case of the AND-joint return period surpasses the trivariate case of the OR-joint return period where the TP, V, DAND=5, 405.93 years, while TP, V, DOR=500.46 years. This comprehensive approach enhances hydrological modeling and decision-making for water resource management and flood mitigation projects.

A Comparison of Copulas Based on the Range Distribution and Its Application

This paper compares some Archimedean Copulas based on the range distribution and its application. We show that the Clayton Copula is better than the other Copula models considered in this paper. Also, Clayton Copula performed better than the regular one-way ANOVA when there were no outliers. However, when the data had outliers, the Clayton Copula performed almost at the same level (power-wise) as the regular one-way ANOVA.&#x0D; &#x0D; &amp;nbsp;

Probabilistic analysis of a sustainable landfill cover considering stress-dependent water retention model and copula-based random fields

Material properties, crucial design parameters in landfill cover systems, can exhibit spatial variability due to non-uniform particle size and uneven compaction, affecting cover performance. Additionally, cover thickness, another key design parameter, can induce stress variation and influence soil-water retention capability. Previous designs always ignored these uncertainties and stress effects, potentially leading to unreasonable design schemes. This study aims to provide design recommendations for a three-layer landfill cover system under heavy rainfall, considering these factors simultaneously by the random finite element method (RFEM). The stress effects are captured using a stress-dependent soil-water retention curve (SDSWRC) and the hysteresis in the SDSWRC is considered in probabilistic analyses. For practical applications, uncertainties in two easily controllable particle size parameters, D10 and D60, are depicted by copula-based cross-correlated random fields. Based on empirical equations, the spatial variability of SDSWRC and water permeability function (WPF) can be further characterised by D10 and D60 random fields. It is found that considering SDSWRC and its spatial variability (e.g., shape) can overestimate percolation by at least 1.7 times compared to deterministic analysis using SWRC. The spatial variability of SDSWRC shape has a more significant impact than spatially variable saturated water permeability on cover system performance. Among candidate copulas, the CClayton copula (survival copula of Clayton copula) yields the most conservative results due to its upper tail dependence. The particle size and thickness of the bottom layer influence the cover system performance more than the upper two layers. A bottom layer with D10≤ 0.009 mm and D60/D10≥ 7.54, and thickness ≥ 0.8 m is recommended to achieve a satisfactory performance level with a 100-year design life.

Jump Risk Contagion and Determinants Driven by COVID-19 in Sino-US Stock Market: An Empirical Analysis from a Dependence Perspective

This paper examines the jump risk contagion between the US and China during the financial crisis driven by COVID-19, and the impact of a series of determinants to detect the transmission mechanisms. Specifically, we employ the ARJI–GARCH model to capture jump behavior and apply the Clayton Copula to construct lower tail jump contagion. Furthermore, we conduct regression analysis on variables related to economic fundamentals and investor behavior to explore the fundamental or pure contagion hypotheses. Our results suggest an increased jump dependence during major catastrophic crises such as COVID-19, but it is closer to a memory-less process. Besides, bilateral trade, similarity in fundamentals and investor behavior are all important determinants, and pure contagion hypothesis exists between the Chinese and US stock markets during the pandemic.

Bayesian parametric estimation based on left-truncated competing risks data under bivariate Clayton copula models

In observational/field studies, competing risks and left-truncation may co-exist, yielding ‘left-truncated competing risks’ settings. Under the assumption of independent competing risks, parametric estimation methods were developed for left-truncated competing risks data. However, competing risks may be dependent in real applications. In this paper, we propose a Bayesian estimator for both independent competing risks and copula-based dependent competing risks models under left-truncation. The simulations show that the Bayesian estimator for the copula-based dependent risks model yields the desired performance when competing risks are dependent. We also comprehensively explore the choice of the prior distributions (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy) and hyperparameters via simulations. Finally, two real datasets are analyzed to demonstrate the proposed estimators.