Bivariate Archimedean Copulas Research Articles

On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class Card of all d-dimensional Archimedean copulas with d≥3 and proves the afore-mentioned implication with respect to conditioning on the first d−1 coordinates. Several properties equivalent to pointwise convergence in Card are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses μC(Lt) and the Kendall distribution function FKd as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators ψ of d-dimensional Archimedean copulas from the perspective of their so-called Williamson measures γ on (0,∞) is then shown to allow not only to derive surprisingly simple expressions for μC(Lt) and FKd in terms of γ and to characterize pointwise convergence in Card by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of γ directly carry over to the corresponding copula Cγ∈Card. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular d-dimensional copulas is dense in Card and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.

Co-occurrence probability of water balance elements in a mountain catchment on the example of the upper Nysa Kłodzka River

Conditions of the formation of key elements of the water balance, such as precipitation and runoff, and relations between them in the mountain catchment area are very complicated, conditioned both by the climatic factor and the physiographic characteristics of the catchment area. The aim of the study is to determine relations between precipitation and runoff in the Kłodzka Valley (KV) located in mountain areas of south-western Poland. Analyzes were based on precipitation in KV and discharges of the Nysa Kłodzka River and its tributaries, recorded in hydrological years 1974–2013. The bivariate Archimedean copulas were used to describe the degree of synchronicity between these variables. The study area shows a considerable variability in the conditions of transformation of precipitation into runoff. It is conditioned both by the pluvial regime and the physical-geographical characteristics of the catchment area. As a result, sub-catchments with diversified hydrological activity and their role in the formation of water resources of the entire KV were identified. Among them, the Biała Lądecka River sub-catchment was found to be the most hydrologically active, and the sub-catchment of Bystrzyca Dusznicka River the most inert, despite e.g. quite similar synchronicity of precipitation compared to the average precipitation in KV. At the same time, the KV rivers are characterized by different types of runoff regime and characteristic of the water balance structure. The methodology presented can be useful in determining dependencies between selected elements of the water balance and evaluation of water resources availability in source areas of mountain rivers.

EVALUATION OF ERA5 AND IMERG PRECIPITATION DATA FOR RISK ASSESSMENT OF WATER CYCLE VARIABLES OF A LARGE RIVER BASIN IN SOUTH ASIA USING SATELLITE DATA AND ARCHIMEDEAN COPULAS

Precipitation as a major water cycle variable influences the occurrences and distribution of terrestrial water storage change (TWSC), evapotranspiration (ET), and river discharge (Q) of a large river basin. However, its relationship with the other water cycle variables using probabilistic dependence structure concept has not been addressed much. Furthermore, precipitation derived from gauge record is plagued by bias due to orography and under-catch. To fill these gaps, bivariate copula and precipitation derived from reanalysis and satellite data were used. In the present study, the basin-wide averages of the precipitation products APHRODITE, ERA5, and IMERG were used as predictors, whereas the areal mean of MOD16 evapotranspiration, GRACE TWSC, and gauge discharge were used as dependent variables (predictants) for the Brahmaputra basin. The bivariate Archimedean copulas were applied to all the pairs of precipitation-TWSC, precipitation-ET and precipitation-Q based on the optimal marginal distributions obtained. Using the best copula for each pair of the variables, the conditional probability was constructed to predict the predictants for different precipitation amounts (5th, 25th, 50th, 75th, and 95th percentiles). The focus of the analysis was on two scenarios of the predictants (i.e.,≤ 5th and ≥ 95th percentiles). The non-exceedance conditional distribution of TWSC, ET, and Q (all predictants ≤ 5th percentile) decreases with precipitation increase. However, the exceedance probability of the predictants (≥ 95th percentile) increases gradually with an increase in precipitation. The results revealed that both ERA5 and IMERG precipitation data could be used to derive probabilistic measures of the water cycle variables in the absence of gauge-based precipitation.

Probabilistic Approach to Precipitation-Runoff Relation in a Mountain Catchment: A Case Study of the Kłodzka Valley in Poland

On the basis of daily precipitation and discharges recorded in 1974–2013 relations between precipitation and runoff in the Kłodzka Valley (KV) in south-western Poland were analyzed. The degree of synchronicity between them was determined using the bivariate Archimedean copulas. This study aims at identifying and then describe in a probabilistic way the precipitation and runoff relations in the area playing an important role in the formation of water resources, but also particularly exposed to flooding. It was found that isolines of the synchronous occurrence of precipitation and total runoff in the Nysa Kłodzka catchment controlled by gauge Kłodzko had a zonal distribution, with the synchronicity values decreasing from south-east to north-west of the study area. This proves that its eastern part is more hydrologically active, compared to the western part, and as such it determines the amount of water resources of the study area. The decrease in synchronicity is influenced by the type and spatial distribution of precipitation, the structure of water supply, and the geological structure of the study area. Moreover, probabilistic methods applied in this research differ from those used in previous research on the hydrology of KV, as we propose using the copula functions. The method presented can be used to evaluate the availability of water resources in areas playing a key role in their formation on different scales.

On convergence of associative copulas and related results

AbstractTriggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulas 𝒞ardifferent notions of convergence - standard uniform convergence, convergence with respect to the metricD1, and so-called weak conditional convergence - coincide, in the current contribution we tackle the natural question, whether the obtained equivalence also holds in the larger class of associative copulas 𝒞a. Building upon the fact that each associative copula can be expressed as (finite or countably infinite) ordinal sum of Archimedean copulas and the minimum copulaMwe show that standard uniform convergence and convergence with respect toD1are indeed equivalent in 𝒞a. It remains an open question whether the equivalence also extends to weak conditional convergence. As by-products of some preliminary steps needed for the proof of the main result we answer two conjectures going back to Durante et al. and show that, in the language of Baire categories, when working withD1a typical associative copula is Archimedean and a typical Archimedean copula is strict.

A new bivariate Archimedean copula with application to the evaluation of VaR

Abstract In this paper, a new generator function is proposed and based on this function a new Archimedean copula is introduced. The new Archimedean copula along with three representatives of Archimedean copula family which are Clayton, Gumbel and Frank copulas are considered as models for the dependence structure between the returns of two stocks. These copula models are used to simulate daily log-returns based on Monte Carlo (MC) method for calculating value at risk (VaR) of the financial portfolio which consists of two market indices, Ford and General Motor Company. The results are compared with the traditional MC simulation method with the bivariate normal assumption as a model of the returns. Based on the backtesting results, describing the dependence structure between the returns by the proposed Archimedean copula provides more reliable results over the considered models in calculating VaR of the studied portfolio.

Lorenz-generated bivariate Archimedean copulas

Abstract A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

Lorenz-Generated Bivariate Archimedean Copulas

An alternative generating mechanism for non-strict bivariate Archimedean copulas via the curve of a positive random variable is proposed. curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these Lorenz copulas, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating curve. For instance, by selecting random variables with lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An alchemy of curves that can be used as general framework to build multiparametric families of copulas is also discussed.

Two-Dimension Monthly River Flow Simulation Using Hierarchical Network-Copula Conditional Models

River flow simulation is required on water resources planning and management. This paper proposes hierarchical network-copula conditional models to generate two-dimension monthly streamflow matrix aiming at simulating flow both on time and space. HNCCMs develop the simulation generator driven by both temporal and spatial covariates conditioned upon values of a set of parameters and hyper parameters which can be addressed from the three-layer hierarchical system. In the first layer, streamflow time series of the station at the most upstream is generated using bivariate Archimedean copulas and river flow space series in each month at stations down a river in sequence is simulated by nested copulas in the second layer. Last, the seasonal characters of the temporal parameters and covariates are detected as well as the spatial ones are detected using the neural network by fitting them into functions to contribute to the downscaling of space series. A case study for the model is carried on the Yellow River of China. This case (1) detects temporal and spatial relationships which illustrate the capacity of catching the seasonal characterize and spatial trend, (2) generates a river flow time series at Huayuankou station as well as (3) simulates a flow space sequence in January picking out best fitted building blocks for the cascade of bivariate copulas, and finally (4) synthesizes two-dimension simulation of monthly river flow. The result illustrates the essentially pragmatic nature of HNCCMs on simulation for this spatiotemporal monthly streamflow which is nonlinear and complex both on time and space.

Probabilistic modelling of drought events in China via 2-dimensional joint copula

Probabilistic modelling of drought events is a significant aspect of water resources management and planning. In this study, popularly applied and several relatively new bivariate Archimedean copulas were employed to derive regional and spatial based copula models to appraise drought risk in mainland China over 1961–2013. Drought duration (Dd), severity (Ds), and peak (Dp), as indicated by Standardized Precipitation Evapotranspiration Index (SPEI), were extracted according to the run theory and fitted with suitable marginal distributions. The maximum likelihood estimation (MLE) and curve fitting method (CFM) were used to estimate the copula parameters of nineteen bivariate Archimedean copulas. Drought probabilities and return periods were analysed based on appropriate bivariate copula in sub-region I–VII and entire mainland China. The goodness-of-fit tests as indicated by the CFM showed that copula NN19 in sub-regions III, IV, V, VI and mainland China, NN20 in sub-region I and NN13 in sub-region VII are the best for modeling drought variables. Bivariate drought probability across mainland China is relatively high, and the highest drought probabilities are found mainly in the Northwestern and Southwestern China. Besides, the result also showed that different sub-regions might suffer varying drought risks. The drought risks as observed in Sub-region III, VI and VII, are significantly greater than other sub-regions. Higher probability of droughts of longer durations in the sub-regions also corresponds to shorter return periods with greater drought severity. These results may imply tremendous challenges for the water resources management in different sub-regions, particularly the Northwestern and Southwestern China.

Copula-based Local Dependence Framework

This paper proposes a novel copula-based local Kendall’s tau framework to uncover richer nonlinear local dependence between two financial return series. This framework nests the concepts of global dependence, tail dependence and local dependence. Closed form solutions of local Kendall’s tau in terms of copula link local dependence with their global dependence structure together, providing a generalized framework for investigating dependence between two return series. We further extend the copula-based local dependence framework to Spearman’s rho. Using this framework, we draw the local Kendall’s tau surfaces in different quadrants for some common used bivariate Archimedean copulas. Finally, we demonstrate the advantages of copula-based local Kendall’s tau relative to global Kendall’s tau with stock market data.

Joint risk of interbasin water transfer and impact of the window size of sampling low flows under environmental change

Constructing a joint distribution of low flows between the donor and recipient basins and analyzing their joint risk are commonly required for implementing interbasin water transfer. In this study, daily streamflow data of bi-basin low flows were sampled at window sizes from 3 to183days by using the annual minimum method. The stationarity of low flows was tested by a change point analysis and non-stationary low flows were reconstructed by using the moving mean method. Three bivariate Archimedean copulas and five common univariate distributions were applied to fit the joint and marginal distributions of bi-basin low flows. Then, by considering the window size of sampling low flows under environmental change, the change in the joint risk of interbasin water transfer was investigated. Results showed that the non-stationarity of low flows in the recipient basin at all window sizes was significant due to the regulation of water reservoirs. The general extreme value distribution was found to fit the marginal distributions of bi-basin low flows. Three Archimedean copulas satisfactorily fitted the joint distribution of bi-basin low flows and then the Frank copula was found to be the comparatively better. The moving mean method differentiated the location parameter of the GEV distribution, but did not differentiate the scale and shape parameters, and the copula parameters. Due to environmental change, in particular the regulation of water reservoirs in the recipient basin, the decrease of the joint synchronous risk of bi-basin water shortage was slight, but those of the synchronous assurance of water transfer from the donor were remarkable. With the enlargement of window size of sampling low flows, both the joint synchronous risk of bi-basin water shortage, and the joint synchronous assurance of water transfer from the donor basin when there was a water shortage in the recipient basin exhibited a decreasing trend, but their changes were with a slight fluctuation, in particular at shorter window sizes. The decreasing magnitude of individual water shortage risk in the recipient basin was significantly greater than that of joint bi-basin water shortage risk at all window sizes. Although the donor basin had much rich water, but its individual risk accounted for more than 25.3% within a window size of 30days provided that its instream flow was maintained. In practical design and management of interbasin water transfer, impacts of the window size of sampling low flows under environmental change on the joint distribution of bi-basin low flows and the individual risk of water shortage in the donor basin should be paid more attention.

Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures

A new class of bivariate distributions is introduced that extends the Generalized Marshall-Olkin distributions of Li and Pellerey (2011). Their dependence structure is studied through the analysis of the copula functions that they induce. These copulas, that include as special cases the Generalized Marshall-Olkin copulas and the Scale Mixture of Marshall-Olkin copulas (see Li, 2009),are obtained through suitable distortions of bivariate Archimedean copulas: this induces asymmetry, and the corresponding Kendall's tau as well as the tail dependence parameters are studied.

İki Boyutlu Arşimedyen Kapulalar İçin Bazı Seçim Kriterlerinin Karşılaştırılması

Commonly, while selecting an appropriate bivariate Archimedean copula function that models data, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and minimum distance (MD) are used as a selection criterion. In this study, the performances of these criteria for selecting copula function are investigated by some simulation studies.

Study of vertical breakwater reliability based on copulas

The reliability of a vertical breakwater is calculated using direct integration methods based on joint density functions. The horizontal and uplifting wave forces on the vertical breakwater can be well fitted by the lognormal and the Gumbel distributions, respectively. The joint distribution of the horizontal and uplifting wave forces is analyzed using different probabilistic distributions, including the bivariate logistic Gumbel distribution, the bivariate lognormal distribution, and three bivariate Archimedean copulas functions constructed with different marginal distributions simultaneously. We use the fully nested copulas to construct multivariate distributions taking into account related variables. Different goodness fitting tests are carried out to determine the best bivariate copula model for wave forces on a vertical breakwater. We show that a bivariate model constructed by Frank copula gives the best reliability analysis, using marginal distributions of Gumbel and lognormal to account for uplifting pressure and horizontal wave force on a vertical breakwater, respectively. The results show that failure probability of the vertical breakwater calculated by multivariate density function is comparable to those by the Joint Committee on Structural Safety methods. As copulas are suitable for constructing a bivariate or multivariate joint distribution, they have great potential in reliability analysis for other coastal structures.

Copula-based risk assessment of drought in Yunnan province, China

Yunnan is one of the provinces which had been frequently and heavily affected by drought disasters in China. Recently, large severe droughts struck Yunnan, caused considerable social, economic and ecological losses. A risk assessment of meteorological drought for Yunnan province is provided in this study. Based on the daily meteorological data of 29 stations during 1960–2010, duration and severity as two major drought characteristics, defined by the runs and the composite meteorological drought index, are abstracted from the observed drought events. Three bivariate Archimedean copulas are employed to construct the joint distributions of the drought characteristics. Based on the error analysis and tail dependence coefficient, the Gumbel–Hougaard copula is selected to analyze spatial distributions of the joint return periods of drought. The results indicate that a high risk is observed in the middle parts and the northeast parts of Yunnan province, while a relative lower drought risk is observed in the northwest of Yunnan province. The probabilistic properties can provide useful information for water resources planning and management.

ON SOME PROPERTIES OF TWO VECTOR-VALUED VAR AND CTE MULTIVARIATE RISK MEASURES FOR ARCHIMEDEAN COPULAS

AbstractWe consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. andDi Bernardino, E. (2013)Journal of Multivariate Analysis,119, 32–46;Cousin, A. andDi Bernardino, E. (2014)Insurance: Mathematics and Economics,55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.

Estimation of multivariate conditional-tail-expectation using Kendall's process

This paper deals with the problem of estimating the multivariate version of the Conditional-Tail-Expectation, proposed by Di Bernardino et al. [(2013), ‘Plug-in Estimation of Level Sets in a Non-Compact Setting with Applications in Multivariable Risk Theory’, ESAIM: Probability and Statistics, (17), 236–256]. We propose a new nonparametric estimator for this multivariate risk-measure, which is essentially based on Kendall's process [Genest and Rivest, (1993), ‘Statistical Inference Procedures for Bivariate Archimedean Copulas’, Journal of American Statistical Association, 88(423), 1034–1043]. Using the central limit theorem for Kendall's process, proved by Barbe et al. [(1996), ‘On Kendall's Process’, Journal of Multivariate Analysis, 58(2), 197–229], we provide a functional central limit theorem for our estimator. We illustrate the practical properties of our nonparametric estimator on simulations and on two real test cases. We also propose a comparison study with the level sets-based estimator introduced in Di Bernardino et al. [(2013), ‘Plug-In Estimation of Level Sets in A Non-Compact Setting with Applications in Multivariable Risk Theory’, ESAIM: Probability and Statistics, (17), 236–256] and with (semi-)parametric approaches.

Bayesian Inference of Pair-Copula Constriction for Multivariate Dependency Modeling of Iran’s Macroeconomic Variables

Shahid Chamran University, Ahvaz, Iran Bayesian inference of pair-copula constriction (PCC) is used for multivariate dependency modeling of Iran’s macroeconomics variables: oil revenue, economic growth, total consumption and investment. These constructions are based on bivariate t-copulas as building blocks and can model the nature of extreme events in bivariate margins individually. The model parameter was estimated based on Markov chain Monte Carlo (MCMC) methods. A MCMC algorithm reveals unconditional as well as conditional independence in Iran’s macroeconomic variables, which can simplify resulting PCC’s for these data. Key words: Monte Carlo Markov Chain Method, pair-copula construction, vine. Introduction Multivariate data usually exhibit a complex pattern of dependency. Methods such as graphical model and Bayesian networks are available to investigate dependency structures in multivariate data. One increasingly popular approach for constructing high dimensional dependency is based on copulas. Copulas are multivariate distribution functions with uniform margins which allow representation of joint distribution functions as a function of marginal distributions and a copula (Sklar, 1959). Copulas are used in various fields of applied sciences, but are most widely used in economics, finance and risk management (Embrechts, et al., 2003; Patton, 2004; Nolte, 2008). The class of copulas for bivariate data is rich in comparison to the one for �−dimensional data with � ≥ 3. Untilrecently, Gaussian and t-copulas or, more M. R. Zadkarami is a associated professor on the Faculty of Mathematics and Computer Sciences in the Statistics Department. Email him at: zadkarami@yahoo.co.uk. O. Chatrabgoun is a PhD student on the Faculty of Mathematics and Computer Sciences in the Statistics Department. Email him at: o-chatrabgoun@phdstu.scu.ac.ir. generally, elliptical copulas, have been used for multivariate data (Frahm, et al., 2003). The generalization of bivariate copulas to multivariate copulas of dimensions larger than 2 is not straightforward, however there is one simple generalization for Archimedean copulas known as exchangeable Archimedean copulas (Frey & McNeil, 2003). It should be noted that not all bivariate Archimedean copulas have a corresponding multivariate exchangeable version (Nelsen, 1999). Approaches for constructing multivariate Archimedean copulas of more than 2 have dimensions been developed by Joe (1997), Embrechts, et al. (2003), Whelan, (2004), McNeil, et al. (2006), Savu and Trede (2006) and McNeil (2007). Joe (1996) and Bedford and Cooke (2001, 2002) constructed flexible higher-dimensional copulas by using only bivariate copulas as building blocks, which they termed vines. Kurowicka and Cooke (2006) discussed Gaussian vine constructions in details. Aas, et al. (2007) first recognized the general construction principle for deriving multivariate copulas; they used more general bivariate copulas than the Gaussian copula and applied these construction methods to financial risk data using more appropriate pair-copulas such as the bivariate t Clayton and Gumbel copulas. According to recent empirical investigations of Berg and Aas (2007) and Fischer, et al. (2007), the vine constructions based on bivariate t-

Risk Assessment of Hydroclimatic Variability on Groundwater Levels in the Manjara Basin Aquifer in India Using Archimedean Copulas

AbstractIn this paper, a bivariate-copula-based methodology is presented to assess the risk associated with hydroclimatic variability on groundwater levels in an unconfined aquifer at the Manjara basin in India. Rank correlation analysis is used to identify the association between the El Nino–Southern Oscillation (ENSO) index, precipitation, and groundwater levels. It is found that the dependencies among the hydroclimatic variable pairs are statistically significant and the dependence structure can be modeled by using bivariate Archimedean copulas. The groundwater level or depth-to-groundwater table (DGWT) in the study region is found to be responsive toward interannual precipitation variations that are influenced by the ENSO phenomenon. For probabilistic representation of hydroclimate variables, various probability distributions are evaluated and it is found that the precipitation and DGWT are best fitted using lognormal and Weibull distributions, respectively, whereas the ENSO index is best fitted using...