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An approach to modelling volatile financial return series using stationary d-vine copula processes combined with Lebesgue-measure-preserving transformations known as v-transforms is proposed. By developing a method of stochastically inverting v-transforms, models are constructed that can describe both stochastic volatility in the magnitude of price movements and serial correlation in their directions. In combination with parametric marginal distributions it is shown that these models can rival and sometimes outperform well-known models in the extended GARCH family. 1

The goal of the present research work is to develop a new family of aggregation operators (AOs) under the fractional orthotriple fuzzy (FOF) information and apply them to MCDM problems. At the start, the Archimedean coupla and co-coupla are extended to handle FOF information, and some operational law for fractional orthotriple fuzzy numbers (FOFNs) based on extended Archimedean copula (EAC) and extended Archimedean co-copula (EACC) is presented. The fractional orthotriple fuzzy Banzhaf Choquet-Copula aggregation operator and fractional orthotriple fuzzy Banzhaf Choquet-Copula geometric operators are defined based on the proposed operational laws of FOFNs. In addition, the modified maximum deviation approach and the Banzhaf function model are created to objectively determine the fuzzy measure (FM) of criteria sets. Finally, based on the defined AOs, the appropriate decision-making procedures are developed. The proposed approaches can effectively overcome the FMs of criteria sets that are subjectively assigned by decision makers, as well as some decision-making problems in which the weights of the criteria are incomplete (completely) unknown, and there is a correlation between all criteria set.

Economic and financial time series data can exhibit nonstationary and nonlinear patterns simultaneously. This paper studies copula-based time series models that capture both patterns. We introduce a procedure where nonstationarity is removed via a filtration, and then the nonlinear temporal dependence in the filtered data is captured via a flexible Markov copula. We propose two estimators of the copula dependence parameters: the parametric (two-step) copula estimator where the marginal distribution of the filtered series is estimated parametrically; and the semiparametric (two-step) copula estimator where the marginal distribution is estimated via a rescaled empirical distribution of the filtered series. We show that the limiting distribution of the parametric copula estimator depends on the nonstationary filtration and the parametric marginal distribution estimation, and may be non-normal. Surprisingly, the limiting distribution of the semiparametric copula estimator using the filtered data is shown to be the same as that without nonstationary filtration, which is normal and free of marginal distribution specification. The simple and robust properties of the semiparametric copula estimators extend to models with misspecified copulas, and facilitate statistical inferences, such as hypothesis testing and model selection tests, on semiparametric copula-based dynamic models in the presence of nonstationarity. Monte Carlo studies and real data applications are presented.

This work presents a derivative-free trust-region algorithm for probability maximization problems. We assume that the probability function is continuously differentiable with Lipschitz continuous gradient, but no derivatives are available. The algorithm explores the particular structure of the probability objective function through models based on copulae. Under reasonable assumptions, the global convergence of the algorithm is analyzed: we prove that all accumulation points of the sequence generated by the algorithm are stationary. The proposed approach is validated by encouraging numerical results on academic and industrial problems.

In this paper, we consider a new family of multivariate copulas described by a sequence of functions, named as AMO copula. The set of AMO copulas corresponds to a class of multivariate shock models with the Archimedean type of dependence. Sufficient conditions on the involved sequence of functions to obtain a multivariate copula are given and the probabilistic structure of the AMO copula is provided. We show that the family of AMO copulas is a generalization of Archimedean and Marshall-Olkin copulas family, and it includes some well-known copulas as specific cases. An alternative method for generating random vectors from AMO copulas via distortion functions is provided. In addition, a singular component along the main diagonal of the AMO copula is also verified. Finally, the tail behaviors of AMO copulas are discussed and some numerical illustrations are provided.

Considering the well-known shuffling operation in x- and in y-direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator S T induced by pairs T = ( T 1 × T 2 ) of measure preserving transformations on ( [ 0 , 1 ] , B ( [ 0 , 1 ] ) , λ ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d ∞ . After deriving some general properties of the set Ω T of all S T -invariant copulas we focus on the situation where T 1 , T 2 are strongly mixing and show that in this case the product copula Π is an extreme point of Ω T . Moreover, motivated by a recent paper by Horanska and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Luroth maps and derive various additional properties of Ω T , including the surprising fact that Ω T contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction.

In this paper we find pointwise best-possible bounds on the set of copulas with a given value of the Spearman's footrule coefficient. We show that the lower bound is always a copula but, unlike the bounds on sets of copulas with a given value of other measures, such as Kendall's tau, Spearman's rho and Blonqvist's beta, the upper bound can be a copula or a proper quasi-copula. We characterised both of these cases.

This paper deals with the problem of characterizing all functions f : [ 0 , 1 ] → R + such that C f ( x , y ) = x y f ( ( 1 − x ) ( 1 − y ) ) , x , y ∈ [ 0 , 1 ] is a bivariate copula. We provide a complete characterization for the two cases: (i) when C f is, in addition, totally positive of order 2 (TP2) and (ii) when f is twice continuously differentiable. In general, the function f need only be twice differentiable Lebesgue almost everywhere as shown by investigating necessary conditions for C f to be a copula. The paper also contains numerous examples illustrating obtained results and connections to known facts from the literature. Moreover, several properties of such copulas are described.

The DEA is a nonparametric method of assessing the efficiency of decision-making units using mathematical programming. The classic DEA model assumes that input and output variables are deterministic. However, there are many applications where the variables are of a stochastic nature. Based on the consideration of input and output levels as random variables, the Stochastic Data Envelopment Analysis (SDEA) was developed. Statistical distributions therefore play a major role in this regard. By considering the dependency between input and output variables, and also their simultaneous dependencies in this study, we have introduced three copula-SCCR models. We used three copulas of Gaussian, Clayton, and Gumbel to estimate the dependence between the random variables with normal distribution. We evaluated the proposed models using real data from 20 bank branches. The results showed that considering stochastic dependency between the inputs or outputs causes different results. A comparison between the Copula-SCCR models and the SCCR models revealed that the efficiency of the DMUs using the Copula-SCCR models differed from the SCCR model by a significant margin of at least 20%.

For a typical insurance portfolio, the claims process for a short period, typically one year, is characterized by observing frequency of claims together with the associated claims severities. The collective risk model describes this portfolio as a random sum of the aggregation of the claim amounts. In the classical framework, for simplicity, the claim frequency and claim severities are assumed to be mutually independent. However, there is a growing interest in relaxing this independence assumption which is more realistic and useful for the practical insurance ratemaking. While the common thread has been capturing the dependence between frequency and aggregate severity within a single period, the work of Oh et al. (2021) provides an interesting extension to the addition of capturing dependence among individual severities. In this paper, we extend these works within a framework where we have a portfolio of microlevel frequencies and severities for multiple years. This allows us to develop a factor copula model framework that captures various types of dependence between claim frequencies and claim severities over multiple years. It is therefore a clear extension of earlier works on one-year dependent frequency-severity models and on random effects model for capturing serial dependence of claims. We focus on the results using a family of elliptical copulas to model the dependence. The paper further describes how to calibrate the proposed model using illustrative claims data arising from a Singapore insurance company. The estimated results provide strong evidences of all forms of dependencies captured by our model.

In this paper, we seek to examine the relationship and dynamic dependence structure between the Australian dollar (AUD), euro (EUR), and the British pound (GBP), expressed in American dollars (USD) using a multivariate fractional cointegration model. Our empirical analysis reveals several important findings. First, the main advantage of this approach is to detect the long-term relationship as well as the short-term dynamics and to represent the interdependence between the variables. We further estimate a multivariate GARCH type model that enables us to examine the dynamic conditional correlations (short-run links) among the considered variables under the effects of long-run interactions and volatility persistence. We determine that the volatility transmission was time-varying and that influence from the crisis. Moreover, the joint distribution is explored using the Gumbel copulas in order to describe the nonlinear structure of dependence between the variables. The empirical results provide evidence of fractional cointegration between the exchanges rates and show long-run causal links between the variables and we find significant bidirectional causal links. In particular, we show a positive dynamic correlation and the dependence structure is represented by the optimal copula coefficient used for measuring the risk spillovers between the exchange rates.

We propose a methodology for synthetic generation of solar irradiance (shortwave flux) and air temperature time series using copula functions. The use of copulas for the simulation gives flexibility to represent the serial stochastic variability of the solar irradiation and the air temperature affecting the photo-voltaic (PV) panel energy output. Moreover, it allows to have more control on the desired properties of the time series, not only in the temporal and cross-dependencies, but also in the marginal distributions. We use mixtures of zero mass adjusted density distributions to assess the nature of solar irradiance, alongside vector generalized linear models for the bivariate time series time marginal distributions. We found that the copula autoregressive methodology used, including the zero mass characteristics of the solar irradiance time series, accurately models the stochastic phenomena. Experimental analysis with observed data substantiates the usage and convenience of the proposed methodology to model solar irradiance time series and solar energy across the northern hemisphere, southern hemisphere and equatorial zones. These results will improve the understanding of the fluctuating nature of solar irradiance and also help to understand the underlying stochastic process of photo-voltaic energy production.

Objectives The original 3-level EQ-5D (EQ-5D-3L) includes 5 dimensions with 3 levels of problems per dimension. Since 2010, a more sensitive version with 5 levels of problems per dimension (EQ-5D-5L) has become available. Population value sets have been developed for both versions of the questionnaire. The objective of this research was to develop a mapping function to link EQ-5D-3L responses to value sets for the EQ-5D-5L. Methods Various algorithms were developed to link EQ-5D-3L and EQ-5D-5L responses using data from an observational study including members of 10 subgroups (N = 3580) who completed both versions of the questionnaire. Nonparametric and ordinal logistic regression models were fit to the data and compared using Akaike’s information criterion (AIC) as well as the mean absolute error and root mean squared error of predictions. Results were contrasted qualitatively and quantitatively with those of an alternative copula-based approach. Results Including indicants of problems for other EQ-5D-3L dimensions as regressors in the modeling yielded the greatest improvement in prediction accuracy. Adding age and gender lowered the AIC without improving predictions, while including a latent factor lowered the AIC further and slightly improved predictive accuracy. Models that conditioned on problems in other EQ-5D-3L dimensions yielded more accurate predictions than the alternative copula-based approach in subgroups defined by age and gender. Conclusion We present novel algorithms to map EQ-5D-3L responses to EQ-5D-5L value sets. The recommended approach is based on an ordinal logistic regression that disregards age and gender and accounts for unobserved heterogeneity using a latent factor.

Based on the random vector ( X + Z , Y + Z ) we study the perturbation C X + Z , Y + Z of the copula C X , Y of the random vector ( X , Y ) when the random noise Z is independent of both X and Y. We also propose a bivariate copula-based modification of the Irwin-Hall distribution and use it to extend the results of Mesiar et al. (2019) and present several examples for illustration.

Vine copula, constructed from bivariate copulas, provides great flexibility in modelling complex high-dimensional dependence. When applied to multi-population mortality modelling, vine copula yields significant improvement over traditional multivariate copulas. In this paper, we propose to capture time-varying features in mortality dependence with dynamic regular vine (R-vine) copula which is built from bivariate copulas with time-varying dependence parameters. We develop two dependence dynamics for R-vine copulas and illustrate the selection and estimation of dynamic R-vine copulas using mortality data from eight populations. The estimated R-vine copulas using the proposed dependence dynamics are shown to yield better goodness of fit than both static and regime-switching vine copulas. We further demonstrate the simulation of mortality paths using dynamic R-vine copulas and examine the impact of vine copula choice on the assessed effectiveness of longevity hedge.

Structural risk assessment is very important for maintaining the safe operations of many complex systems. However, there are two major requirements that still need to be addressed, namely the correlation among multiple factors and high interpretability to produce a trustworthy result. For the first challenge on correlation, the Copula model is applied to quantitatively measure the correlation among the data gathered from multiple sources. For the second challenge on interpretability, the Belief Rule Base (BRB) model is applied since it is essentially a white-box approach that can provide good interpretability by its transparent inferencing and decision-making process. With this, a novel approach is proposed by combining the Copula model and BRB, namely Copula-BRB. Moreover, two frameworks are separately designed as well. The first is a knowledge-driven inferencing framework when experts’ knowledge is available but with limited data. The second is a data-driven optimization framework when there is a large quantity of data. Correspondingly, a numerical case with only limited data and another practical case with 1000 sets of data are studied to explain and validate the two frameworks of the Copula-BRB approach, respectively. The numerical case is mainly used to explain the knowledge-driven Copula-BRB inferencing framework, and the practical case is conducted in a comparative fashion where three Copula functions are used, namely Clayton, Frank, and Gumbel, and the results are compared with the Support Vector Machine (SVM) and conventional BRB (without consideration of the attribute correlation). The results of the two case studies show that, by using the Copula-BRB which incorporates the attribute correlation into the conventional BRB, superior performance has been achieved in comparison with SVM and BRB. A detailed exploration of how this superior performance concerning outputs in intervals has been conducted as well. Ministry of Education (MOE) Nanyang Technological University The Start-Up Grant at Nanyang Technological University, Singapore (No. 04INS000423C120), the Ministry of Education Grants, Singapore (No. 04MNP002126C120; No. 04MNP000279C120), and the National Natural Science Foundation of China (Grant Nos. 71601180 and 61903108) are acknowledged for their financial support of this research. This work was also supported by the National Key Research and Development Program of China under Grant 2017YFB120700, and the BNRist Program under Grants BNR2019TD01009.

In the modeling of multivariate extreme risks, the tail dependence and the heavy tailedness are the two key factors. Heavy tailedness are usually defined through the regular variation. Tail dependence can be modeled by copulas with the so-called tail order property. In this paper, we propose a new risk measure called the Joint Expected Shortfall (JES) as an alternative of quantifying extreme risks. The JES, which can be viewed as a consolidation of both Expected Shortfall (ES) and Marginal Expected Shortfall (MES) risk measures, has the desirable property of measuring risk by jointly capturing both tail dependence and heavy tailedness. The asymptotic analysis of JES is conducted to provide a simple and transparent way of studying the interplay between tail dependence and heavy tailedness. Various examples are presented to illuminate our results. In particular, risk measures such as ES and MES that ignore the joint effect of dependence and heavy tailedness may severely underestimate the underlying risk.

Several extensions of the family of (bivariate) Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas) are considered. Some of them are well-known from the literature, others have recently been suggested (copulas based on quadratic constructions, based on some forms of convexity, and polynomial copulas). For each of these extensions we analyze which properties of EFGM copulas are preserved (or even improved) and which are (partly) lost. Such properties can be structural (order theoretical or topological) in nature, or algebraic (symmetry or being a polynomial) or analytic (absolute continuity). Other examples are forms of convexity, quadrant dependence, and symmetry with respect to copula transformations. The last group of properties considered here is related to some dependence parameters.

First, we recall the properties of the curved section of a copula that is specified by an automorphism of the unit interval. Then, inspired by the operation of diagonal splicing of two copulas, we develop the curved splicing operation, which essentially glues together different restrictions of two copulas that share the same curved section. In order to exploit this operation in practice, we propose two new classes of semilinear copulas. Applying the curved splicing operation to members of these two classes leads to two other classes of semilinear copulas.

Multiple disasters induced by hurricanes such as floods, strong winds, and heavy rainfall have always been a significant threat to coastal infrastructures. However, few studies have attempted to model the dependence between hazards and study its impact on multi-hazard risks—these cognitive deficiencies inevitably affect the control of risks. This paper addresses these difficulties by first using the copulas to capture the dependence between hurricane-induced floods and strong winds based on synthetic multi-hazard events. Moreover, another two types of dependence, regression-based and mutually independent, are constructed to compare the impact of different dependence behaviors on the facility’s failure. Second, the frequency of concurrent events and a scenario-based fragility function are convolved to quantify the multi-hazard risk of storage tanks. Subsequently, the annualized failure risk (AFR) and the mean return period of failure (MRPF) are used as two metrics for the multi-hazard risk. The case study results demonstrate that the BB1 copula can accurately describe the change in dependence across the full range of the multivariate distribution, avoiding overestimation or underestimation of equipment failure risks. The application of risk zoning provides decision-makers with more intuitive and convenient tools for planning and layout, highlighting the practicability of the proposed multi-hazard risk assessment method.

We study different redundancy mechanisms in coherent systems with possibly dependent heterogeneous components. The dependence is modelled through copulas. We use distortion functions to obtain general results for different redundancy procedures. This model includes the popular active redundancy and minimal repair procedures. The purpose is to determine where these redundant components should be located in the system, according to its structure. We study series and parallel systems in detail. In the first case (series systems), it is well known that active and minimal repair redundancies should be assigned to the weakest component. However, we show that, surprisingly, this is not always the case for all the redundancy mechanisms. Moreover, we give a condition on the redundancy mechanism to get this expected property. Similar results are obtained for parallel systems. In these systems one can think that the best option is to assign the redundancy to the strongest component. However, we prove that this is not always the case. We include results for other system structures as well and we show that these properties also depend on the copula (dependence structure).

In this paper, we develop two adaptive approaches for detecting the signal source in a bivariate process when a shift occurs in the location vector or the scale matrix or both. The proposed method capitalises the notion of Sklar’s principle of expressing any multivariate joint distribution in terms of univariate marginal-distribution functions and a copula, which represents the dependence structure between the variables. Motivated by this, we recommend monitoring the two marginal distributions and the copula function simultaneously using appropriate nonparametric (distribution-free) test statistics. At each stage of Phase-II monitoring, we adopt the permutation method for computing the individual p-values and derive the plotting statistics of our proposed schemes combining suitable transforms of the three p-values of the component testing. We establish the in-control robustness of the proposed surveillance plans and compare them with two competitors in terms of run length properties. Performance of the proposed schemes in detecting a correct out-of-control signal is as good or better than some existing charting schemes for bivariate process monitoring. The novelty of our proposed technique lies in the fact that it indigenously helps in identifying the component(s) responsible for the signal, which is not straightforward with the traditional schemes for surveillance of a bivariate process. Numerical results substantiate that the proposed procedure performs significantly better than its competitors in many cases. Also, we investigate the percentage of correct diagnosis of a signal via the proposed charting schemes. Nowadays, in monitoring and control of smooth service operations, the use of quality monitoring has increased than ever before, but the problem and data structures become more complicated in the Industry 4.0 era. We analyse two real case studies, one in the context of monitoring the response time and service quality in a call centre and the other related to the inspection of product quality, to illustrate the application of the proposed schemes.

In general, common cause failures (CCFs) have been modeled with the assumption that components within the same group are symmetric. This assumption reduces the number of parameters required for the CCF probability estimation and allows us to use a parametric model, such as the alpha factor model. Although there are various asymmetric conditions in nuclear power plants (NPPs) to be addressed, the traditional CCF models are limited to symmetric conditions. Therefore, this paper proposes the copula-based CCF model to deal with asymmetric as well as symmetric CCFs. Once a joint distribution between the components is constructed using copulas, the proposed model is able to provide the probability of common cause basic events (CCBEs) by formulating a system of equations without symmetry assumptions. In addition, Bayesian inferences for the parameters of the marginal and copula distributions are introduced and Markov Chain Monte Carlo (MCMC) algorithms are employed to sample from the posterior distribution. Three example cases using simulated data, including asymmetry conditions in total failure probabilities and/or dependencies, are illustrated. Consequently, the copula-based CCF model provides appropriate estimates of CCFs for asymmetric conditions. This paper also discusses the limitations and notes on the proposed method.

A novel multi-criteria decision making approach based on an adaptive copula Bayesian network (CBN) model is proposed to effectively handle complex dependence problems under uncertainty. Specifically, the Bayesian network is used to merge various criteria in a model and graphically describe cause-effect relationships. The copula is incorporated to construct joint distributions of variables by specifying marginal distributions and copula functions separately. Regarding the practical value, the constructed model can adjust to changeable conditions to provide adaptive suggestions in view of diverse disciplines. The effectiveness of the proposed approach is verified in a case study about choosing the most suitable location of the pedestrian overhead bridge (POB) to install lift facilities in Singapore. Firstly, three criteria and ten relevant influential factors concerning the sustainable aspect are derived from a combination of expert knowledge and statistical data. Then, a proper CBN model is developed under the consideration of these determined criteria and factors, aiming to predict the constructability index for alternative locations statistically. Finally, the correlation analysis and CBN inference are performed to evaluate and identify the ideal location in a data-driven manner. Furthermore, results from the proposed CBN-based approach are compared against the traditional Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) to illustrate its advantages in representing dependency, modeling uncertainty, fusing information, and reducing subjectivity. Ministry of Education (MOE) Nanyang Technological University The Ministry of Education Tier 1 Grant, Singapore (No. 04MNP000279C120 and the Start-Up Grant at Nanyang Technological University, Singapore (No. 04INS000423C120) are acknowledged for their financial support of this research.

In this work, we study the problem of (sub)copula estimation via continuity of the Sklar's correspondence. One benefit of this approach is that the estimator can be obtained from that of the corresponding (joint) distribution function via plug-in method. Additional proof is not required. Our approach is to naturally embed the space of subcopulas into the space of distribution functions. This allows us to consider two common modes of convergence, namely, the uniform convergence and the weak convergence on the space of (sub)copulas. Since these modes of convergence are well-studied, we will be able to combine our results with existing results in the same way that is done for distribution functions. Instead of proving these results directly, however, we will apply two common distances characterizing these modes of convergence, namely, the Chebyshev distance and the Levy distance. By using distances, the rate of convergence is also known. Topological properties of the space of subcopulas based on these two distances are also compared and contrasted with that of previously defined distances.

The multiple-grammar model (Iwasaki 2015) was proposed based on the observation that the same speaker produces radically different forms of language in the spoken and written modes of language use. The model relies strongly on recent developments in usage-based grammars and cognitive linguistics, and considers speakers’ actual experience as the key to developing different grammatical systems in two distinct linguistic environments. It also acknowledges that a speaker may develop different systems (e.g. grammars) in different varieties, such as conversations and public speeches, within the same mode of language use. In order to understand how such grammatical varieties are structured, we focus in this paper on copula (desu) constructions in Japanese, which include a typical predicative copula construction (PCC), e.g. hon desu ‘(it) is a book,’ an unusual non-predicative copula construction (NPCC), e.g. soshite desu ne ‘then, you know,’ and the formulaic copula constructions (FCC), e.g. soo desu ne ‘that’s right.’ These constructions are distributed differently not only across speech activities, but also across individual speakers. This paper suggests that speakers’ differential experiences with the constructions used in different speech activities may be the cause of such variability.

In this paper we introduce some new copulas emerging from shock models. It was shown in [21] that reflected maxmin copulas (RMM for short) are not just some specific singular copulas; they contain many important absolutely continuous copulas including the negative quadrant dependent “half” of the Eyraud-Farlie-Gumbel-Morgenstern class. The main goal of this paper is to develop the RMM copulas with dependent endogenous shocks and give evidence that RMM copulas may exhibit some characteristics better than the original maxmin copulas (MM for short): (1) An important evidence for that is the iteration procedure of the RMM transformation which we prove to be always convergent and we give many properties of it that are useful in applications. (2) Using this result we find also the limit of the iteration procedure of the MM transformation thus answering a question proposed in [10] . (3) We give the multivariate dependent RMM copula that compares to the MM version given in [10] . In all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to Marshall copulas where symmetric linking functions are used.

Constructing new copulas by transforming a given copula is necessary to enhance the flexibility of the existing copula family. Among various copula transformations, the Khoudraji transformation (K-transformation), described in [16] , provides a promising way to create asymmetric copulas. In this study, we completely characterize the invariance property of the K-transformation and discuss how the K-transformation changes measures of association or asymmetry of a copula. In particular, this study explores the identifiability issue of Khoudraji-transformed copula families. Some conditions for guaranteeing identifiable transformed copula families are provided.

In De Baets et al. [6] the authors define, for a copula C, an idempotent transformation D [ C ] which, in some cases, generates new copulas. In this note we give a (negative) answer to a question posed in [6] , showing that D [ C ] is not necessarily a copula when C is an absolutely continuous Archimedean copula.

In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: Montes et al. (2015) [13] . The main tools we develop in order to do so are: (1) a theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) p-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) p-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladic and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) p-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual σ-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.

This article provides an alternative to High-Dimensional Model Representation using a Copula approximation of an unknown functional form. We apply our methodology in the context of an extensive Monte Carlo study and to a sample of large US commercial banks. In the Monte Carlo experiment, the approximations errors of the Copula approach are small and behave randomly. In our empirical application, we find that the Copula Approximation performs much better, in terms of Bayes factors for model comparison, compared to High-Dimensional Model Representation, which, in turn, provides better results when compared with standard flexible functional forms, like the translog, the minflex Laurent, and the Generalized Leontief, or a Multilayer Perceptron. Moreover, the choice of approximation has significant implications for productivity and its components (returns to scale, technical inefficiency, technical change, and efficiency change).

This paper addresses the problem of quantification and propagation of uncertainties associated with dependence modeling when data for characterizing probability models are limited. Practically, the system inputs are often assumed to be mutually independent or correlated by a multivariate Gaussian distribution. However, this subjective assumption may introduce bias in the response estimate if the real dependence structure deviates from this assumption. In this work, we overcome this limitation by introducing a flexible copula dependence model to capture complex dependencies. A hierarchical Bayesian multimodel approach is proposed to quantify uncertainty in dependence model-form and model parameters that result from small data sets. This approach begins by identifying, through Bayesian multimodel inference, a set of candidate marginal models and their corresponding model probabilities, and then estimating the uncertainty in the copula-based dependence structure, which is conditional on the marginals and their parameters. The overall uncertainties integrating marginals and copulas are probabilistically represented by an ensemble of multivariate candidate densities. A novel importance sampling reweighting approach is proposed to efficiently propagate the overall uncertainties through a computational model. Through an example studying the influence of constituent properties on the out-of-plane properties of transversely isotropic E-glass fiber composites, we show that the composite property with copula-based dependence model converges to the true estimate as data set size increases, while an independence or arbitrary Gaussian correlation assumption leads to a biased estimate.

This paper presents a mathematical model for the prediction of the probabilities of reverse power flow exceeding predefined critical thresholds at feed-in points of a distribution network. The parametric prediction model is based on hourly forecasts of global horizontal irradiation and uses copulas, a tool for modeling the joint probability distribution of two or more strongly correlated random variables with non-Gaussian (marginal) distributions. The model is used for determining the joint distribution of forecasts of global horizontal irradiation and measured solar power supply at given feed-in points, where respective sample datasets were provided by Deutscher Wetterdienst and the N-ERGIE Netz GmbH. It is shown that the fitted model replicates important characteristics of the data such as the corresponding marginal densities. The validation results highlight strong performance of the proposed model. The copula-based model enables to predict the distribution of solar power supply conditioned on the forecasts of global horizontal irradiation, thus anticipating great fluctuations in the distribution network.

This study endeavours to shed some light on the Contagion risk in the Vietnamese banking system. In so doing, we analyse the contagion risk through stock returns on listed commercial banks by employing non-parametric and Copula approaches. A rich set of empirical approaches are employed, including non-parametric (Chi-plots, Kendall-plots) and parametric Copula estimations to define the dependence structure of pairs of daily returns, balanced by a total of 36 copulas with 17,456 observations over the period from July 2006 to September 2017. Our results show that the risk of each individual bank may transmit to other banks through stock returns, which are reflected in their price information. The results also suggest existence of contagion risk and strong dependency in the structure of stock returns of banks under analysis. As a consequence, to avoid negative returns for the portfolio, careful diversification is required while investing in the Vietnamese banking sector, when showing a Clayton relationship (left-tail dependency). Our findings have profound implications for investors, policymakers and authorities responsible for financial stability.

Starting from a bivariate copula family, this paper investigates the existence of a Markov process whose temporal dependence is modeled by the given copula family. Due to that the transition function plays a core role for constructing a Markov process, a transition function should be defined from a copula family. For this purpose, the modified partial Dini derivatives of a bivariate copula are defined and applied for defining transition probabilities, and some properties of the modified partial Dini derivatives are proved. A necessary and sufficient condition for the family of the defined transition probabilities to be a transition function is provided. Given a bivariate copula family, a sufficient condition for the existence of a Markov process is provided, where the Markov process has a transition function generated by the modified partial Dini derivatives of the bivariate copula family and the temporal dependence of the Markov process is modeled by the given copula family. The resulting Markov process is named as the copula-based Markov process. Moreover, under some assumptions the consistency of the bivariate copula family under the ∗ product operation is necessary and sufficient for the existence of a Markov process. In terms of copulas, some criteria are provided for a copula-based Markov process to be path right-continuous with left limits or path continuous, and a necessary and sufficient condition for a time-homogeneous copula-based Markov process to be a Feller process is obtained. It is interesting that a Markov process with the transition function generated by the modified partial Dini derivatives of FGM copulas is not a Feller process. Finally, paths of some typical copula-based Markov processes are simulated to show the importance of fitting the copula method into the framework of stochastic processes.

This paper proposes an asymptotically distribution-free test for copulas with dynamic marginal distributions, such as GARCH and ARMA processes. The test is based on the empirical copula process with parametrically estimated marginal distributions. By applying the Khmaladze (1982, 1988, 1993) martingale transformation method, the transformed empirical process converges to a standard Gaussian process, so the resulting test statistics are asymptotically distribution-free. Monte Carlo simulations show that the test performs well in finite samples. An empirical application to test copulas between EUR/USD and GBP/USD exchange rates is provided.

The paper introduces a Generalized-Convex-Sum-Transformation of aggregation functions. It has the form of a transformation of aggregation functions by monotone systems of functions. A special case of the proposed Generalized-Convex-Sum-Transformation is the well-known ⁎-product, also called the Darsow product of copulas. Similarly, our approach covers Choquet integrals with respect to capacities induced by the considered aggregation function. The paper offers basic definitions and some properties of the mentioned transformation. Various examples illustrating the transformation are presented. The paper also gives two alternative transformations of aggregation functions under which the dimension of the transformed aggregation functions is higher than that of the original one. Interestingly, if a copula is transformed, under some conditions put on the monotone systems of functions, the transformed aggregation function is again a copula.

Ambiguity refers to a fogginess inducing some lack of precision in variables, parameters or magnitudes in models whose aim is to represent real phenomena or better that are strictly linked to the human feelings and beliefs. It has recently become an usual model assumption in several stream of literature. Here we focus on multivariate models affected by ambiguity and provide a rigorous modellization of the main ingredients causing a multidimensional fuzzification. We introduce a conditional fuzzified model, where a certain level of uncertainty affects the set of univariate margins individually taken and also an unconditional model where the ambiguity involves the dependency structure as well. Both these models, i.e. the conditional and the unconditional fuzzy copula model, are compared and their convergence is discussed. Finally a pricing application of these multidimensional fuzzy models, based on Sugeno measures, is proposed.

We introduce a new method for constructing fuzzy implications based on arbitrary two copulas and some fuzzy implication. Our approach covers two well-known constructions of fuzzy implications based on an arbitrary single copula. We also discuss some properties of the proposed method and provide several interesting examples.

Copula models have become popular in different applications, including modeling shocks, in view of their ability to describe better the dependence concepts in stochastic systems. The class of maxmin copulas was recently introduced by Omladic and Ružic [21] . It extends the well known classes of Marshall–Olkin and Marshall copulas by allowing the external shocks to have different effects on the two components of the system. By a reflection (flip) in one of the variables we introduce a new class of bivariate copulas called reflected maxmin (RMM) copulas. We explore their properties and show that symmetric RMM copulas relate to general RMM copulas similarly as semilinear copulas relate to Marshall copulas. We transfer that relation also to maxmin copulas. We also characterize possible diagonal functions of symmetric RMM copulas.

Nonparametric moment-based goodness-of-fit tests are developed for the family of chi-square copulas of arbitrary dimensions. This class of dependence models allows for tail asymmetries and contains the family of multivariate normal copulas as a special case. The proposed tests are based on two rank correlation coefficients whose population versions are equal, up to a monotone transformation, when the underlying dependence structure is a chi-square copula. The test statistics are computed from natural rank-based estimations of these two correlation coefficients and their large-sample distributions under the null hypothesis of a chi-square copula are derived; the validity of a parametric bootstrap procedure for the computation of p-values is formally established as well. Particular attention is given to tests for the families of normal and centered chi-square copulas. The simulations that are reported indicate that the new tests are reliable alternatives to those based on the empirical copula, both in the bivariate and multivariate cases. The usefulness of the introduced methodology is illustrated on the five-dimensional Nutrient dataset.

When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We compute the sharp bound of asymmetry measure $\mu_\infty$, the most important of the asymmetry measures, for the family of Marshall copulas and the family of maxmin copulas, which both equal to $\frac{4}{27}\ (\approx 0.148)$. One should compare this bound to the one for the class of PQD copulas to which they belong, which is $3-2\sqrt{2}\ \approx 0.172)$, and to the general bound for all copulas that is $\frac13$. Furthermore, we give the sharp bound of the same asymmetry measure for RMM copulas which is $3-2\sqrt{2}$, compared to the same bound for NQD copulas, where they belong, which is $\sqrt{5}-2\ (\approx 0.236)$. One of our main results is also the statistical interpretation of shocks in a given model at which the maximal asymmetry measure bound is attained. These interpretations for the three families studied are illustrated by examples that should be helpful to practitioners when choosing the model for their data., Comment: The latest version (V.4) contains a correction in Theorem 3.1 and Remark 3.3 compared to the printed version in the journal and to the previous version on arXiv. Functions $P_{\lambda}$ in the earlier version of Theorem 3.1 are not copulas as claimed. We wish to thank Professor Piotr Jaworski for pointing out the fact. (31 pages, 14 figures)

Consider a vector ( X , Y ) that describes the failure times of two non-independent components of a system. Assuming that the first component has survived up to a given time t > 0 , i.e., assuming X > t , one can define the corresponding residual lifetime under different assumptions on the failure of the second component, having lifetime Y . In particular, one can observe that the second component has not failed before a time s ≥ 0 (maybe different from t ), thus the conditioned residual lifetime X ˜ t = [ X − t | X > t , Y > s ] can be considered, or one cannot observe Y , and in this case the conditioned residual lifetime X t = [ X − t | X > t ] has to be studied. This note deals with conditions on the survival copula of ( X , Y ) such that X ˜ t and X t are comparable according to the main reliability stochastic orders. Similar conditions, based on the connecting copula of ( X , Y ) , are described also for the conditioned inactivity times X ˜ t = [ t − X | X ≤ t , Y ≤ s ] and X t = [ t − X | X ≤ t ] .

Estimation of a change point is a classical statistical problem in sequential analysis and process control. For binomial time series, the existing maximum likelihood estimators (MLEs) for a change point are limited to independent observations. If the independence assumption is violated, the MLEs substantially lose their efficiency, and a likelihood function provides a poor fit to the data. A novel change point estimator is proposed under a copula-based Markov chain model for serially dependent observations. The main novelty is the adaptation of a three-state copula model, consisting of the in-control state, out-of-control state, and transition state. Under this model, a MLE is proposed with the aid of profile likelihood. A parametric bootstrap method is adopted to compute a confidence set for the unknown change point. The simulation studies show that the proposed MLE is more efficient than the existing estimators when serial dependence in observations are specified by the model. The proposed method is illustrated by the jewelry manufacturing data, where the proposed model gives an improved fit.

Copula theory is concerned with defining dependence structures given appropriate marginal distributions. Probabilistic sensitivity analysis is concerned with quantifying the strength of the dependence among the output of a simulator and the uncertain simulator inputs. In this work, we investigate the connection between these two families of methods. We define four classes of sensitivity measures based on the distance between the empirical copula and the product copula. We discuss the new classes in the light of transformation invariance and Renyi’s postulate D of dependence measures. The connection is constructive: the new classes extend the current definition of sensitivity measures and one gains an of understanding which sensitivity measures in use are, in fact, copula-based. Also a set of new visualization tools can be obtained. These tools ease the communication of results to the modeler and provide insights not only on statistical dependence but also on the partial behavior of the output as a function of the inputs. Application to the benchmark simulator for sensitivity analysis concludes the work.

We consider the winning probability relation associated with a set of (real-valued) random variables. Its computation requires knowledge of the marginal cumulative distribution functions of the random variables and of the bivariate cumulative distribution function of any couple of these random variables. Here, the copulas underlying these bivariate cumulative distribution functions are assumed to be identical and to belong to the parametric family of Frank copulas, although this most likely does not represent the real dependence and might even not be feasible at all; the Frank copula parameter should thus be regarded as a parameter of the method presented. This winning probability relation is then exploited to establish a strict partial order on the set of random variables. This is realized by computing an appropriate α-cut that yields a cycle-free crisp relation, resulting in a strict partial order upon computing its transitive closure. For any given Frank copula, we focus on finding the lowest possible value of α, called (minimal) cutting level, such that cutting the winning probability relation strictly above that value results in a crisp relation that is free from cycles of length m, with m ∈ { 3 , 4 , … } , irrespective of the marginal cumulative distribution functions of the random variables. We are able to give closed-form expressions for these cutting levels: (1) for all m when the copula is either the independence copula or one of the two Frechet-Hoeffding bounds; (2) for any Frank copula when m = 4 or when m tends to infinity. The relationship between the cutting level, the cycle length and the Frank copula parameter is discussed.

Data-driven based prognostics typically requires sufficient run-to-failure training units in order to learn degradation characteristics of engineering components or products without the need of understanding the physics-based degradation mechanisms. With insufficient training units, however, the model form learned from the training units may be inaccurate, which could result in large remaining useful life (RUL) prediction errors for the actual test units. A recently proposed copula-based sampling method does not assume any degradation model form, but builds a set of statistical correlation models for the RUL prediction, which shows high RUL prediction accuracy with sufficient training units. This paper proposes an enhanced copula-based method to address the instability issue of the method when dealing with insufficient training units. In particular, the sampling part for the RUL prediction in the original time domain is replaced by an analytical formulation in the standard uniform space with multiple copulas so that the stability issue can be addressed. Furthermore, a simplified version of the RUL prediction is proposed with extremely high efficiency. Effectiveness of the enhanced method is demonstrated by two case studies: one for resistance degradation of lead-acid batteries and another for capacity degradation of lithium-ion batteries.

Tail dependence of crude oil price returns between four major benchmark markets are analyzed through the lenses of nonparametric copula models. This paper illustrates that nonparametric copula is flexible to incorporate important empirical patterns of tail dependence and provides better goodness-of-fit to the data than the optimal parametric copula. Estimation results show that the level and the structure of tail dependence of crude oil returns vary significantly depending on data frequency and the time period covered.

In recent years, copula models are being used in all areas of human endeavors including energy, environment, social, natural and physical sciences. Copulas are the most powerful tools that can model dependent structures between various complex correlated variables. In this paper, we specifically examine the development of copula models and their applications in the areas of energy, fuel cells, forestry and environmental sciences. It reviews the latest literature on the types of copula models, including Gumbel, Clayton, Frank, Gaussian, vine, and the theoretical development of a mixture of bivariate and multivariate copula distributions, in terms of both static and dynamic applications. A comparative review of literature is done using ARMA, DCC, and GARCH models relative to copulas.

Finding parametric models that accurately describe the dependence structure of observed data is a central task in the analysis of time series. Classical frequency domain methods provide a popular set of tools for fitting and diagnostics of time series models, but their applicability is seriously impacted by the limitations of covariances as a measure of dependence. Motivated by recent developments of frequency domain methods that are based on copulas instead of covariances, we propose a novel graphical tool that allows to access the quality of time series models for describing dependencies that go beyond linearity. We provide a thorough theoretical justification of our approach and show in simulations that it can successfully distinguish between subtle differences of time series dynamics, including non-linear dynamics which result from GARCH and EGARCH models. We also demonstrate the utility of the proposed tools through an application to modeling returns of the S&P 500 stock market index., paper (20 pages) and online supplement (9 pages), 15 figures