Archimedean Copulas Research Articles

MODELING THE BENEFITS OF A MARRIAGE REVERSE ANNUITY CONTRACT WITH DEPENDENCY ASSUMPTIONS USING ARCHIMEDEAN COPULA

Social security benefits may not be enough for retirement. Equity release products like marriage reverse annuities can boost retirement income for older couples. Marriage reverse annuity’s contract convert all or part of the real estate value of elderly spouses while they are living (joint life status) or after one partner dies (last survivor status). Since husband and wife face the same death risk, the chance of death between spouses is believed to be dependent for realism. Thus, copula models the future dependency model of a husband and wife. Sklar's theorem states that copulas link bivariate distribution and marginal cumulative functions. One of the most common copulas is Archimedean copula. Clayton, Gumbel, and Frank are Archimedean copula that will be used in this investigation. The Indonesian Mortality Table IV data is used to obtain the marginal distribution of the male and female which will then be used to construct copulas (Clayton, Gumbel, and Frank) that combine two marginal distributions into a joint distribution. The marginal distribution of Indonesian Mortality Table IV is uncertain, hence Canonical Maximum Likelihood parameter estimation is utilized to estimate the parameter of copulas. Multiple-state models depict the marriage reverse annuity model for joint life and last survivor status. The probability structure is based on Sklar's theorem and copula survival function. The contract benefits calculation utilizing copulas (Clayton, Gumbel, and Frank) shows that joint life status benefits are higher than last survivor status. Joint life status uses the dependence assumption with Frank's copula to calculate the smallest annual benefit value of a marriage reverse annuity contract, while last survivor status uses the independence assumption (without copula).

Multivariate copula-based conditional quantiles: analytic higher-order moments and ratio estimation approaches

ABSTRACT In this paper, we aim to modify multivariate copula-based conditional quantiles for a targeted random variable, given multiple random variables attaining their respective quantiles. Specifically, we propose two modifications through the Cornish–Fisher expansion, one of which involves its analytic higher-order unconditional moments. The second one accounts for its higher-order conditional moments estimated using a ratio estimation method. By considering multivariate elliptical and Archimedean copulas with Johnson's SU margins, our simulation study shows that this method provides more accurate conditional moment estimators compared to the naive method. They result in expanded conditional quantile estimators, whose efficiency is relatively better than their unexpanded versions, particularly at lower and higher quantile levels under a stronger (tail) dependence assumption. A much higher efficiency is gained when the first modification is performed. These findings are validated using an empirical study based on cryptocurrency return data and a comparative analysis against the existing estimation approaches.

Generalized likelihood ratio method for stochastic models with uniform random numbers as inputs

We propose a new unbiased stochastic gradient estimator for a family of stochastic models driven by uniform random numbers as inputs. Dropping the requirement that the tails of the density of the input random variables decay smoothly, the estimator extends the applicability of the generalized likelihood ratio (GLR) method. We demonstrate the new estimator for several general classes of input random variates, including independent inverse transform random variates and dependent input random variables governed by an Archimedean copula. We show how the new estimator works in settings such as density estimation, and we illustrate applications to credit risk derivatives. Numerical experiments substantiate broad applicability and flexibility in dealing with discontinuities in the sample performance.

Exploring a new two-parameter Archimedean copula: the Gumbel-Joe copula

In this article, we introduce a novel Archimedean copula constructed from a unique strict generator function. It can be described as a two-parameter unification of the well-established Gumbel-Barnett and Joe copulas. The first part is devoted to its formulation, as well as those of the corresponding density, the conditional copula, and the Kendall distribution function. Additionally, graphical representations are included to elucidate their shape behavior under various parameter configurations. In a second part, we examine some of its notable properties, with emphasis on correlation characteristics. Practical applications are discussed in the last part. In particular, we employ the maximum likelihood estimation method to determine the unknown parameters involved based on data and conduct a simulation study to demonstrate the effectiveness of this approach. Additionally, we analyze a dataset to provide practical illustrations of copula behavior and potential.

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.

Stochastic ordering results on extreme order statistics from dependent samples with Archimedean copula

This paper considers parallel and series systems with heterogeneous components having dependent exponential lifetimes. The underlying dependence is assumed to be Archimedean and the component lifetimes are supposed to be connected according to an Archimedean copula. Sufficient conditions are found to dominate a parallel system with heterogenous exponential components, with respect to the dispersive order, by another parallel system with homogenous exponential components where the dependence structure between lifetimes of components is the same. We also compare two series systems (and two parallel systems) with general one-parameter dependent components and with respect to the usual stochastic ordering. Examples are given to illustrate the theoretical findings.

Multivariate analysis of compound hail, wind and rainfall extremes in Alberta's hail alley

Hailstorms are severe weather events with the potential for devastating impacts. The consequences can be significantly worsened when hail events are accompanied by strong winds, intensifying both hail momentum and damage to property sidings and windows. Additionally, rainfall extremes during hailstorms can disrupt the drainage systems, potentially leading to flash flooding. Therefore, understanding the inter-dependencies and joint behaviour of these hazards is crucial for developing effective risk mitigation strategies. In this study, we conduct a multivariate probabilistic assessment of concurrent hail, wind, and rainfall extremes over the Alberta's “hail alley” using radar and ground-based observations. The analysis comprehensively explores individual hazards, as well as bivariate and trivariate scenarios using a vine copula approach. We quantify individual, conditional, and joint return periods (JRPs) for the various scenarios. Findings indicate that in both wind-driven hail and hail-rainfall extreme hazards, the joint occurrences based on JRP, can be underestimated by 20% and 70% when assuming independence, respectively, which has substantial implications for risk assessment and management, as well as infrastructure design and maintenance. The analysis of the trivariate case suggests the potential for the concurrent occurrence of multiple hazards in the region. Furthermore, results show that Archimedean copula families outperform elliptical copulas in simulating extreme variables related to compound events associated with hailstorms. The study stresses the importance of assessing the joint behaviour of these hazard components in hailstorms, with the objective of mitigating potential impacts on vulnerable regions.

Drought risk management in agriculture: A copula perspective on crop diversification

AbstractDrought events are a major cause of large crop yield losses with implications for food security and farmers’ incomes. Growing multiple crops simultaneously during a cropping season is a well‐known on‐farm risk management strategy to cope with these drought risks. However, the effectiveness of this crop diversification under different severity levels of drought and how this effectiveness is influenced by the crop composition is unclear. This article provides new methodological and empirical insights to assess the effectiveness of such diversification, in particular to cope with extreme drought. We apply and evaluate nested Archimedean copulas and elliptical copulas to assess simultaneous farm‐level yield losses of different cash crops in German agriculture (winter wheat, winter barley, winter rapeseed, sugar beet, and grain maize) under different drought severity levels (N = 249,756; regionally pooled farm‐level crop‐yield pairs, 1995–2019). We show that on‐farm crop diversification contributes to cope with drought risks, but its effectiveness varies considerably across regions, crop pairs, and drought severity. Our results underline that cropping system diversification alone is often not sufficient to cope with drought risks, but that the right crop combinations are needed. For example, during a severe drought (one in 20 years event), 26.4% of farmers in eastern Germany suffered simultaneous yield losses of at least 20% in winter wheat and winter barley, while 19.1% of farmers in eastern Germany suffered simultaneous yield losses of at least 20% in winter wheat and sugar beet. Farmers should therefore be encouraged to grow crops with more diverse phenological requirements throughout the year.

Time‐varying copula‐based compound flood risk assessment of extreme rainfall and high water level under a non‐stationary environment

AbstractQuantifying flood risk depends on accurate probability estimation, which is challenging due to non‐stationarity and the combined effects of multiple factors in a changing environment. The threat of compound flood risks may spread from coastal areas to inland basins, which have received less attention. In this study, a framework based on time‐varying copulas was introduced for the treatment of compound flood risk and bivariate design in non‐stationary environments. Archimedean copulas were developed to diagnose the non‐stationary trends of flood risk. Return periods, average annual reliabilities, and bivariate designs were estimated. Model uncertainty was analyzed by comparing the results for stationary and non‐stationary conditions. The case study investigated the extreme rainfall and water level series from the Qinhuai River Basin and the Yangtze River in China. The results showed that marginal distributions and correlations are non‐stationary in all bivariate combinations. Ignoring composite effects may lead to inappropriate quantification of flood risk. Excluding non‐stationarity may lead to risk over or underestimation. It showed the limitations of the 1‐day scale and quantified the uncertainty of non‐stationary models. This study provided a flood risk assessment framework in a changing environment and a risk‐based design technique, which is essential for climate change adaptation and water management.

Stochastic comparison of parallel systems with heterogeneous dependent exponential components

Let X=(X1,…,Xn) and Y=(Y1,…,Yn) be two random vectors with common Archimedean copula with generator function ϕ, where, for i=1,…,n, Xi is an exponential random variable with hazard rate λi and Yi is an exponential random variable with hazard rate λ. In this paper we prove that under some sufficient conditions on the function ϕ, the largest order statistic corresponding to X is larger than that of Y according to the dispersive ordering and hazard rate ordering. The new results generalized the results in Dykstra et al. (1997) and Khaledi and Kochar (2000). We show that the new results can be applied to some well known Archimedean copulas.

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.

On multivariate orderings of some general ordered random vectors

Ordered random vectors are frequently encountered in many problems. The generalized order statistics (GOSs) and sequential order statistics (SOSs) are two general models for ordered random vectors. However, these two models do not capture the dependency structures that may be present in the underlying random variables. In this paper, we study the developed sequential order statistics (D-SOSs) and developed generalized order statistics (D-GOSs) models that incorporate dependency structures among ordered random vectors. We then study various univariate and multivariate ordering properties of D-SOS and D-GOS models under Archimedean copula. We develop corresponding results for both one-sample and two-sample situations. We also present some simulational results and a real data analysis for illustrative purpose.

Probabilistic optimal power flow computation for power grid including correlated wind sources

AbstractThis paper sets out to develop an efficient probabilistic optimal power flow (POPF) algorithm to assess the influence of wind power on power grid. Given a set of wind data at multiple sites, their marginal distributions are fitted by a newly developed generalized Johnson system, whose parameters are specified by a percentile matching method. The correlation of wind speeds is characterized by a flexible Liouville copula, which allows to model the asymmetric dependence structure. In order to improve the efficiency for solving POPF problem, a lattice sampling method is developed to generate wind samples at multiple sites, and a logistic mixture model is proposed to fit distributions of POPF outputs. Finally, case studies are performed, the generalized Johnson system is compared with Weibull distribution and the original Johnson system for fitting wind samples, Liouville copula is compared against Archimedean copula for modelling correlated wind samples, and lattice sampling method is compared with Sobol sequence and Latin hypercube sampling for solving POPF problem on IEEE 118‐bus system, the results indicate the higher accuracy of the proposed methods for recovering the joint cumulative distribution function of correlated wind samples, as well as the higher efficiency for calculating statistical information of POPF outputs.

Distribution of the Joint Survival Function of an Archimedean Copula

Suppose a random vector [Formula: see text] with values in the unit cube has a joint survival function: [Formula: see text] given by an Archimedean copula [Formula: see text] with generator [Formula: see text], a smooth decreasing convex function such that [Formula: see text]. In this article, we provide a formula for the distribution of [Formula: see text] where [Formula: see text] is an independent copy of [Formula: see text] and a method to simulate values from the distribution of Z in the bivariate case, that is, when d = 2. The case d > 2 does not seem to be tractable. As an application, we show how our result can be used to compute the limiting covariance of the empirical Kendall process corresponding to [Formula: see text]. AMS Subject Classification: 62H05

Orderings of extremes among dependent extended Weibull random variables

Abstract In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$ , where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$ , for $i=1,\ldots, n$ , which are coupled by Archimedean copulas having different generators. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$ , in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of [5] (Barmalzan, G., Ayat, S.M., Balakrishnan, N., & Roozegar, R. (2020). Stochastic comparisons of series and parallel systems with dependent heterogeneous extended exponential components under Archimedean copula. Journal of Computational and Applied Mathematics380: Article No. 112965).

Parametric dependence between random vectors via copula-based divergence measures

This article proposes copula-based dependence quantification between multiple groups of random variables of possibly different sizes via the family of Φ-divergences. An axiomatic framework for this purpose is provided, after which we focus on the absolutely continuous setting assuming copula densities exist. We consider parametric and semi-parametric frameworks, discuss estimation procedures, and report on asymptotic properties of the proposed estimators. In particular, we first concentrate on a Gaussian copula approach yielding explicit and attractive dependence coefficients for specific choices of Φ, which are more amenable for estimation. Next, general parametric copula families are considered, with special attention to nested Archimedean copulas, being a natural choice for dependence modelling of random vectors. The results are illustrated by means of examples. Simulations and a real-world application on financial data are provided as well.

A new approach for the assessment of coastal flooding risk. Application in Rhodes island, Greece

A directional, copula-based, Monte Carlo approach is proposed for the quantitative assessment of a new definition of coastal flooding risk, namely a probability-based index due to extreme combinations of wave conditions and sea levels, which could be a key element for coastal zone planning and management. This approach differs from the standard practice, which focuses on the extreme events and studies either their specific return values or their worst-case scenarios. The adopted approach incorporates mean wave direction as a covariate into the probabilistic representation of sea states, and via the use of a variety of probabilistic and statistical tools, represents a total of five stochastic variables, namely the significant wave height, the peak wave period, the mean wave direction, the storm surge, and tide. Two directional, alternative methods have been developed; a trivariate copula- and a bivariate copula-based method, that seemed to produce quite similar results. Four types of copulas have been examined for their goodness-of-fit, i.e., three Archimedean copulas, namely, the Clayton, the Gumbel, and the Frank copulas, as well as the elliptical Gaussian copula. A comparative analysis of the two methods is performed aiming to point out their pros, as well as their accuracy and complexity issues. The application of the proposed method to the natural beaches of the Rhodes Island in Greece, in combination with a recently developed formula for wave run-up, highlighted the most crucial parameters affecting the coastal flooding risk in the study area.

Diversifying and hedging REIT portfolios with cryptocurrencies: Evidence from global and regional REIT indices

Hedging the financial risk of portfolios of securitized real estate assets is daunting because of the unique nature of the underlying assets and because no direct market exists to trade on property-related derivatives. In this paper, we conduct a global study of the diversification and risk mitigation benefits of cryptocurrencies for REIT stocks. Specifically, we examine the dynamic relationship between Bitcoin and REIT indices obtained from multiple countries and regions of the world. We also explore whether portfolios of REIT stocks that are hedged with Bitcoin futures obtain significant improvement in portfolio risk mitigation and return performance. Furthermore, given that Bitcoin and REIT stocks are prone to large volatility swings, we examine the nature of the tail dependences in the multivariate distributions of our REIT indices universe and Bitcoin using Archimedean copula functions. Lastly, we compare the market risk of Bitcoin-hedged REIT portfolios to the unhedged portfolios of REIT stocks using value-at-risk. We find region-specific benefits from using Bitcoin to hedge REIT exposure. Furthermore, we show that using Bitcoin to dynamically hedged REIT portfolios had minimal impact on the bivariate and multivariate joint distributions of REIT portfolios. We also find evidence of region-specific reductions in the market risk exposure of dynamically hedged REIT portfolios relative to unhedged and naively hedged REIT portfolios. We posit that Bitcoin could improve the mean-variance frontiers of REIT investors in countries with higher market friction and lower global integration.

Agricultural drought risk assessment in Southern Plateau and Hills using multi threshold run theory

Agricultural drought in this study is characterized and assessed using the Standardized Soil Moisture Index (SSMI). Employing run theory with three thresholds, SSMI identifies grid-wise drought events, quantifying them in terms of peak, frequency, duration and intensity. To evaluate SSMI's performance in assessing drought risk, a comparison is made with the Standardized Precipitation Evapotranspiration Index (SPEI). Spatially, frequency aligns with duration for both indices, indicating regions with longer drought events also experience more frequent occurrences. Meteorological droughts in SPH from 1991 to 2020 are found to be more intense, frequent, longer-lasting, but less severe. Spatially heterogeneous drought events are identified by both indices, with SSMI proving a reliable indicator of drought risk, similar to SPEI. Archimedean copula functions reveal high joint occurrences of various drought characteristics, suggesting significant correlations. The results show that the joint occurrences of duration and peak, intensity and frequency, and duration and peak are all high. This insight is crucial for enhancing drought risk management and developing more effective warning systems.

A bivariate survival model for events with dependent failure times based on Archimedean copula functions. Application case: A sample of HIV patients

This paper proposes a bivariate survival model for dependent failure times based on copula functions of the Archimedean family and the mean cumulative function for non-recurrent events of different types (MCFR ̅E) and uses it to estimate the probability of survival from the occurrence of events of different types on the same HIV/AIDS patient. The copula functions evaluate the dependence structure between the failure times of the events experienced by the same patient throughout their follow-up period, and the MCFR ̅E generates the marginal survival function for each event. The marginal function is a nonparametric estimator that gives the same estimated survival probability as the Kaplan-Meier estimator if the failure times of the different types of events are independent. If each patient experiences at least one event, a subset of them generates a compound event that affects the estimated probability of survival. The results show that the traditionally estimated survival probabilities are biased if dependent failure times are treated as independent.