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The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively. /// Dans cet article on décrit les propriétés de la copule t, avec particulière attention envers la dépendance des valeurs extrêmes. Exploitant la représentation de la loi multivariée t par un mélange de Gaussiennes, on construit deux nouveaux types de copule: une version biaisée (skewed t copula) et une version permettant une majeure hétérogénéité dans la modélisation des observations dépendantes (grouped t copula). Deux autres types de copule sont ensuite construits à l'aide de la théorie des valeurs extrêmes. L'une est la copule limite de la loi des maxima de chaque composante d'un vecteur aléatoire avec distribution t (t extreme value copula), l'autre est la copule limite des observations d'un vecteur bivarié obéissant à une loi t, conditionnées a être en dessous d'un certain seuil commun, qu'on baisse progressivement (t lower tail copula). En ce qui concerne les applications pratiques, ces deux dernières copules peuvent être approximées par d'autres copules plus simples et connues, comme celle de Gumbel et celle de Clayton.

This paper proposes a robust methodology for displaying two-dimensional environmental contour lines based on measured ocean wave data at two offshore sites. For implementing the robust environmental contour lines methodology, we propose the use of bivariate kernel density estimation with smoothed cross-validation bandwidth selection. The environmental contours obtained by using the proposed robust methodology have been compared with those obtained by using the Gumbel copula transformation method and another recently published method, and the effectiveness and superiority of our proposed robust methodology have been clearly substantiated. The research results in this paper demonstrate that our proposed robust methodology can be utilized as an effective tool for predicting the long-term extreme dynamic responses of ocean engineering structures.

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

Multivariate hydrological series become nonstationary under the changing environment. In this paper, a new method is proposed to study the change in the dependence structure between runoff and sediment sequences. First, a moving cut transfer entropy is proposed to detect the sudden change point in the dependence structure between runoff and sediment. Then, the Gumbel copula with a new parameter estimation method (MEE) is employed to construct the dependence structure of runoff and sediment before or after the change point for situation with small samples. The new method is applied to the annual runoff and sediment discharge series in Xiliugou River, China. It is found that the dependence structure between runoff and sediment changed abruptly in 2000. The joint probabilistic characteristics of the annual runoff and sediment discharge during the period of 1960–1999 are built based on the values of RMSEs and AICs. The joint distribution of the annual runoff and sediment discharge during the period of 2000–2016 is constructed by the Gumbel copula with MEE. Moreover, the synchronous encounter probability of runoff and sediment decreases from 0.76 to 0.63, and the asynchronous encounter probability rises from 0.24 to 0.37, after the runoff and sediment discharge series presents a significant change in the dependence structure.

In recent decades, floods have become a major global concern. In Niamey (Niger) in West Africa, flooding is primarily caused by the significant increase in surface runoff resulting from heavy rainfall occuring between July and September in the upstream river basins of the three major tributaries of the Middle Niger River (Sirba, Gorouol ad Dargol catchments). While the Sirba is empirically considered as the largest driver to flooding in Niamey, its contribution have not been precisely established. This study analyzes the influence of these tributaries on the Niger River discharges at Niamey during the rainy season, with a particular focus on the Sirba River basin. Daily annual maximum discharge (AMAX) data from 1990 to 2022 timeseries are used as inputs to various statistical analyses, including trend analyses, change point detection, concordance analysis and flood dependency assessment. The results reveal a significant change point in 2009 and increasing trends between the Sirba and Niger River stations. The flood propagation time delay varies from 1 to 4 days between the upstream river basins tributaries and Niamey station, with a strong concordance in peak discharges, particularly dominant with the Sirba River. The Dynamic Time Warping (DTW) and the Gumbel copula analyses highlighted the significant control of the Sirba River basin on flooding in Niamey, while also highlighting the important roles played by other tributaries. These findings are crucial for improving flood prevention and further refine urban flood management strategies in Niamey and other cities globally, affected by fluvial floods.

This study uses a hybrid model of the exponential generalised auto-regressive conditional heteroscedasticity (eGARCH)-extreme value theory (EVT)-Gumbel copula model to investigate the dependence structure between Bitcoin and the South African Rand, and quantify the portfolio risk of an equally weighted portfolio. The Gumbel copula, an extreme value copula, is preferred due to its versatile ability to capture various tail dependence structures. To model marginals, firstly, the eGARCH(1, 1) model is fitted to the growth rate data. Secondly, a mixture model featuring the generalised Pareto distribution (GPD) and the Gaussian kernel is fitted to the standardised residuals from an eGARCH(1, 1) model. The GPD is fitted to the tails while the Gaussian kernel is used in the central parts of the data set. The Gumbel copula parameter is estimated to be α=1.007, implying that the two currencies are independent. At 90%, 95%, and 99% levels of confidence, the portfolio’s diversification effects (DE) quantities using value at risk (VaR) and expected shortfall (ES) show that there is evidence of a reduction in losses (diversification benefits) in the portfolio compared to the risk of the simple sum of single assets. These results can be used by fund managers, risk practitioners, and investors to decide on diversification strategies that reduce their risk exposure.

When performing validation studies on diagnostic classification procedures, one or more biomarkers are typically measured in individuals. Some of these biomarkers may provide better information; moreover, more than one biomarker may be significant and may exhibit dependence between them. This proposal intends to estimate the Area Under the Receiver Operating  Characteristic Curve  (AUC)  for classifying individuals in a screening study. We analyze the dependence between the results of the tests by means of copula-type dependence (using FGM and Gumbel-Barnett copula functions), and studying the respective AUC under this type of dependence. Three different dependence-level values were evaluated for each copula function considered. In most of the reviewed literature, the authors assume a normal model to represent the performance of the biomarkers used for clinical diagnosis. There are situations in which assuming normality is not possible because that model is not suitable for one or both biomarkers. The proposed statistical model does not depend on some distributional assumption for the biomarkers used for diagnosis procedure, and additionally, it is not necessary to observe a strong or moderate linear dependence between them.

A univariate analysis that relies solely on precipitation data in low flow frequency analysis is a technique to express meteorological drought, so it is limited to analyzing the characteristics of hydrological drought related to available water resources. In addition, if the data for the model calibration are insufficient, the uncertainty of a single variable limits the construction of a reliable model. To improve this problem, a frequency analysis was performed by constructing a bivariate copula model as a multivariate model with a high correlation between variables targeting reservoir inflows. The methodology utilizes the theory of runs to identify low flow events, establishing a threshold based on the mandatory regional water supply plan, and determining the low flow duration and cumulative water deficit. The Gumbel copula function, effective in capturing correlations between hydrological variables, was applied to derive a joint bivariate probability distribution, facilitating the calculation of combined low flow event return periods. This study compared low flow frequencies at Soyanggang dam (’74–’22) and Chungju dam (’86–’22), which are in the same Han River basin but have different capacities and water demands, using a bivariate copula model. The top four extreme low flow events for the two adjacent dam basins did not occur in the same year and, in the years of the extreme low flow events at one of the two dam basins, there was an insignificant magnitude at the remaining dam basin. This result is noteworthy because it shows that the possibility of extreme low flow events appearing simultaneously in both watersheds is not as high as expected. The operational efficiency can be improved by setting the coordinated operation rules of the two reservoirs using the copula dependency structure.

This paper aims to develop a new family of bivariate distributions for modelling different types of claims and their associated costs jointly in a flexible manner. The proposed bivariate distributions can be viewed as a continuous copula distribution paired with two marginals based on composite distributions. For expository purposes, the details of one of the proposed bivarite composite distributions is provided. The dependence measures for the resulting bivariate copula-based composite distribution are studied, and its fitting is compared with other bivariate composite distributions and existing bivariate distributions. The parameters of the proposed bivariate composite model are estimated via the inference functions for margins (IFM) method. The suitability of the proposed bivariate distribution is examined using two real-world insurance datasets, namely the motor third-party liability (MTPL) insurance dataset and Danish fire insurance dataset.

Prediction of water shortage losses is of great importance for water resources management. A new mathematical expression of water shortage loss was proposed in order to describe the random uncertainty and economic attributes of water resources. Then, Gumbel copula with a new method of parameter estimation was introduced to model the joint probabilistic characteristics for water supply and water use in situations when sufficient data is unavailable. The new parameter estimation method requires only the minimum and maximum values of two variables. The improved Gumbel copula was proved to be reliable based on the RMSEs (root mean square error) and AICs (Akaike information criterion), statistical tests and upper tail dependence tests. The potential water shortage losses for all the districts of Tianjin were predicated. The water shortage loss in the Urban district is highest (7.02 billion CNY), followed by the new district of Binhai and Wuqing district, while those in the Baodi district and Ji County are very small. Highlights A new mathematical expression of water shortage loss was proposed in order to describe the random uncertainty and economic attributes of water resources. Gumbel copula with a new method of parameter estimation was introduced to model the joint probabilistic characteristics for water supply and water use in situations when sufficient data is unavailable. The Gumbel copula was proved to be reliable based on the RMSEs (Root mean square error) and AICs (Akaike information criterion), statistical tests and upper tail dependence tests. The potential water shortage losses for all the districts of Tianjin were predicated.