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Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with continuous latent variables, and computing inference using the resulting augmented posterior. To evaluate this, we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis-Hastings step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas can be employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions, we estimate 16-dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

Bias correcting and downscaling climate model simulations requires reconstructing spatial and intervariable dependences of the observations. However, the existing univariate bias correction methods often fail to account for such dependences. While the multivariate bias correction methods have been developed to address this issue, they do not consistently outperform the univariate methods due to various assumptions. In this study, using 20 state-of-the-art coupled general circulation models (GCMs) daily mean, maximum and minimum temperature (T mean , T max and T min ) from the Coupled Model Intercomparison Project phase 6 (CMIP6), we comprehensively evaluated the Super Resolution Deep Residual Network (SRDRN) deep learning model for climate downscaling and bias correction. The SRDRN model sequentially stacked 20 GCMs with single or multiple input-output channels, so that the biases can be efficiently removed based on the relative relations among different GCMs against observations, and the intervariable dependences can be retained for multivariate bias correction. It corrected biases in spatial dependences by deeply extracting spatial features and making adjustments for daily simulations according to observations. For univariate SRDRN, it considerably reduced larger biases of T mean in space, time, as well as extremes compared to the quantile delta mapping (QDM) approach. For multivariate SRDRN, it performed better than the dynamic Optimal Transport Correction (dOTC) method and reduced greater biases of T max and T min but also reproduced intervariable dependences of the observations, where QDM and dOTC showed unrealistic artifacts (T max < T min ). Additional studies on the deep learning-based approach may bring climate model bias correction and downscaling to the next level.

Statistics in sports plays a key role in predicting winning strategies and providing objective performance indicators. Despite the growing interest in recent years in using statistical methodologies in this field, less emphasis has been given to the multivariate approach. This work aims at using the Bayesian networks to model the joint distribution of a set of indicators of players’ performances in basketball in order to discover the set of their probabilistic relationships as well as the main determinants affecting the player’s winning percentage. From a methodological point of view, the interest is to define a suitable model for non-Gaussian data, relaxing the strong assumption on normal distribution in favour of Gaussian copula. Through the estimated Bayesian network, we discovered many interesting dependence relationships, providing a scientific validation of some known results mainly based on experience. At last, some scenarios of interest have been simulated to understand the main determinants that contribute to rising in the number of won games by a player.

Many optimization problems in probabilistic combinatorics and mass transportation impose fixed marginal constraints. A natural and open question in this field is to determine all possible distributions of the sum of random variables with given marginal distributions; the notion of joint mixability is introduced to address this question. A tuple of univariate distributions is said to be jointly mixable if there exist random variables, with respective distributions, such that their sum is a constant. We obtain necessary and sufficient conditions for the joint mixability of some classes of distributions, including uniform distributions, distributions with monotone densities, distributions with unimodal-symmetric densities, and elliptical distributions with the same characteristic generator. Joint mixability is directly connected to many open questions on the optimization of convex functions and probabilistic inequalities with marginal constraints. The results obtained in this paper can be applied to find extreme scenarios on risk aggregation under model uncertainty at the level of dependence.

The modified Dagum distribution is a highly versatile statistical model, and it is included in several important parametric families of distributions, with applications, such as economics and public health. In this paper, we introduce a multivariate version of the modified Dagum distribution and deduce some of its sub-models to address specific analytical needs. We use two different approaches to derive the joint probability density function for the proposed distribution. Also, we derive the joint cumulative distribution function through the traditional method and the Clayton copula methods. In addition, we explore and discuss some statistical properties, including the multivariate dependence. Further, we use the maximum likelihood method to estimate the unknown parameters and the associated confidence interval. Finally, we apply the proposed model to analyze some real data sets, including a protein consumption data set and a warranty policy data set, for demonstrative purposes. The marginals of the proposed model fit the data sets quite well, and the results demonstrate the model’s effectiveness in modeling the proposed data.

Using recent results concerning non-uniqueness of the center of the mix for completely mixable probability distributions, we obtain the following result: For each d ∈ N and each non-empty bounded Borel set B ⊂ R d , there exists a d -dimensional probability distribution μ satisfying the following: For each n ≥ 3 and each probability distribution ν on B , there exist d -dimensional random vectors X ν , 1 , X ν , 2 , ⋯ , X ν , n such that 1 n ( X ν , 1 + X ν , 2 + ⋯ + X ν , n ) ∼ ν and X ν , i ∼ μ for i = 1 , 2 , ⋯ , n . We also show that the assumption regarding the boundedness of the set B cannot be completely omitted, but it can be substantially weakened.

Many modern structural systems usually consist of multiple failure modes. One failure mode may affect other failure modes due to the same working environment and random input variables, meaning that these failure modes are not independent. The assumption of independence among failure modes simplifies the calculation of reliability of structural systems, but it adds approximation error. Different from the dependence between two failure modes, the dependence of multiple failure modes is more complicated. In this paper, on the basis of vine copula function, which is a flexible tool to describe the multivariate dependence, the dependence of multiple failure modes of the structural systems is mainly studied and analyzed. Then Rosenblatt transformation and Monte Carlo simulation method are utilized to evaluate the reliability of structural systems. Furthermore, in order to research the importance of failure modes, two indices combined with empirical copula functions are extended, which can quantitatively measure the importance of failure modes. The multiple failure modes can be ranked based on proposed importance analysis, and the ones that have greater influence on the system can be found out, thus simplifying the system analysis. Finally, in order to confirm the applicability and rationality of the proposed method, an engineering case about a mechanism system with five failure modes is presented.

Reservoirs operation alters the natural flow regime at downstream site and thus has a great impact on the design flood values. The general framework of flood regional composition and Equivalent Frequency Regional Composition (EFRC) method are currently used to calculate design floods at downstream site while considering the impact of the upstream reservoirs. However, this EFRC method deems perfect correlation between peak floods that occurred at one sub-basin and downstream site, which implicitly assumes that the rainfall and the land surface process are uniformly distributed for various sub-basins. In this study, the Conditional Expectation Regional Composition (CERC) method and Most Likely Regional Composition (MLRC) method based on copula function are proposed and developed under the flood regional composition framework. The proposed methods (i.e., CERC and MLRC) are tested and compared with the EFRC method in the Shuibuya-Geheyan-Gaobazhou cascade reservoirs located at Qingjiang River basin, a tributary of Yangtze River in China. Design flood values of the Gaobazhou reservoir site are estimated under the impact of upstream cascade reservoirs, respectively. Results show that design peak discharges at the Gaobazhou dam site have been significantly reduced due to the impact of upstream reservoir regulation. The EFRC method, not taking the actual dependence of floods occurred at various sub-basins into account; as a consequence, it yields an under-or overestimation of the risk that is associated with a given event in hydrological design. The proposed methods with stronger statistical basis can better capture the actual spatial correlation of flood events occurred at various sub-basins, and the estimated design flood values are more reasonable than the currently used EFRC method. The MLRC method is recommended for design flood estimation in the cascade reservoirs since its composition is unique and easy to implement.

A multivariate spatial sampling design that uses spatial vine copulas is presented that aims to simultaneously reduce the prediction uncertainty of multiple variables by selecting additional sampling locations based on the multivariate relationship between variables, the spatial configuration of existing locations and the values of the observations at those locations. Novel aspects of the methodology include the development of optimal designs that use spatial vine copulas to estimate prediction uncertainty and, additionally, use transformation methods for dimension reduction to model multivariate spatial dependence. Spatial vine copulas capture non-linear spatial dependence within variables, whilst a chained transformation that uses non-linear principal component analysis captures the non-linear multivariate dependence between variables. The proposed design methodology is applied to two environmental case studies. Performance of the proposed methodology is evaluated through partial redesigns of the original spatial designs. The first application is a soil contamination example that demonstrates the ability of the proposed methodology to address spatial non-linearity in the data. The second application is a forest biomass study that highlights the strength of the methodology in incorporating non-linear multivariate dependence into the design.

The concept of complete mixability is relevant to some problems of optimal couplings with important applications in quantitative risk management. In this paper we prove new properties of the set of completely mixable distributions, including a completeness and a decomposition theorem. We also show that distributions with a concave density and radially symmetric distributions are completely mixable.