@article { ISI:000355774000005,
	title = {Structured factor copula models: Theory, inference and computation},
	journal = {Journal of Multivariate Analysis},
	volume = {138},
	number = {SI},
	year = {2015},
	month = {JUN},
	pages = {53-73},
	publisher = {Elsevier Inc},
	type = {Article},
	abstract = {In factor copula models for multivariate data, dependence is explained via one or several common factors. These models are flexible in handling tail dependence and asymmetry with parsimonious dependence structures. We propose two structured factor copula models for the case where variables can be split into non-overlapping groups such that there is homogeneous dependence within each group. A typical example of such variables occurs for stock returns from different sectors. The structured models inherit most of dependence properties derived for common factor copula models. With appropriate numerical methods, efficient estimation of dependence parameters is possible for data sets with over 100 variables. We apply the structured factor copula models to analyze a financial data set, and compare with other copula models for tail inference. Using model-based interval estimates, we find that some commonly used risk measures may not be well discriminated by copula models, but tail-weighted dependence measures can discriminate copula models with different dependence and tail properties. (C) 2014 Elsevier Inc. All rights reserved.},
	keywords = {Bi-factor model, Conditional independence, Dependence measure, Factor analysis, Tail asymmetry, Tail dependence, Truncated vine},
	issn = {0047-259X},
	doi = {10.1016/j.jmva.2014.11.002},
	author = {Krupskii, Pavel and Joe, Harry}
}
@article { ISI:000346333600012,
	title = {Tail-weighted measures of dependence},
	journal = {Journal of Applied Statistics},
	volume = {42},
	number = {3},
	year = {2015},
	month = {MAR 4},
	pages = {614-629},
	publisher = {Taylor \& Francis Ltd},
	type = {Article},
	abstract = {Multivariate copula models are commonly used in place of Gaussian dependence models when plots of the data suggest tail dependence and tail asymmetry. In these cases, it is useful to have simple statistics to summarize the strength of dependence in different joint tails. Measures of monotone association such as Kendall{\textquoteright}s tau and Spearman{\textquoteright}s rho are insufficient to distinguish commonly used parametric bivariate families with different tail properties. We propose lower and upper tail-weighted bivariate measures of dependence as additional scalar measures to distinguish bivariate copulas with roughly the same overall monotone dependence. These measures allow the efficient estimation of strength of dependence in the joint tails and can be used as a guide for selection of bivariate linking copulas in vine and factor models as well as for assessing the adequacy of fit of multivariate copula models. We apply the tail-weighted measures of dependence to a financial data set and show that the measures better discriminate models with different tail properties compared to commonly used risk measures - the portfolio value-at-risk and conditional tail expectation.},
	keywords = {62H20, Copula, Dependence measure, factor model, intermediate tail dependence, Tail asymmetry, Tail dependence},
	issn = {0266-4763},
	doi = {10.1080/02664763.2014.980787},
	author = {Krupskii, Pavel and Joe, Harry}
}
@article { ISI:000334819300003,
	title = {Relations between hidden regular variation and the tail order of copulas},
	journal = {Journal of Applied Probability},
	volume = {51},
	number = {1},
	year = {2014},
	month = {MAR},
	pages = {37-57},
	publisher = {Applied Probability Trust},
	type = {Article},
	abstract = {We study the relations between the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics. Multivariate regular variation (MRV) and HRV deal with extremal dependence of random vectors with Pareto-like univariate margins. Alternatively, if one uses a copula to model the dependence structure of a random vector then the upper exponent and tail order functions can be used to capture the extremal dependence structure. After defining upper exponent functions on a series of subcones, we establish the relation between the tail order of a copula and the tail indexes for MRV and HRV. We show that upper exponent functions of a copula and intensity measures of MRV/HRV can be represented by each other, and the upper exponent function on subcones can be expressed by a Pickands-type integral representation. Finally, a mixture model is given with the mixing random vector leading to the finite-directional measure in a product-measure representation of HRV intensity measures.},
	keywords = {intermediate tail dependence, Multivariate regular variation, Tail dependence, tail order function, upper exponent function},
	issn = {0021-9002},
	doi = {10.1017/S0021900200010068},
	author = {Hua, Lei and Joe, Harry and Li, Haijun}
}
@article { ISI:000322290500006,
	title = {Factor copula models for multivariate data},
	journal = {Journal of Multivariate Analysis},
	volume = {120},
	year = {2013},
	month = {SEP},
	pages = {85-101},
	publisher = {Elsevier Inc},
	type = {Article},
	abstract = {General conditional independence models ford observed variables, in terms of p latent variables, are presented in terms of bivariate copulas that link observed data to latent variables. The representation is called a factor copula model and the classical multivariate normal model with a correlation matrix having a factor structure is a special case. Dependence and tail properties of the model are obtained. The factor copula model can handle multivariate data with tail dependence and tail asymmetry, properties that the multivariate normal copula does not possess. It is a good choice for modeling high-dimensional data as a parametric form can be specified to have O(d) dependence parameters instead of O(d(2)) parameters. Data examples show that, based on the Akaike information criterion, the factor copula model provides a good fit to financial return data, in comparison with related truncated vine copula models. (C) 2013 Elsevier Inc. All rights reserved.},
	keywords = {Conditional independence, Factor analysis, Pair-copula construction, Partial correlation, Tail asymmetry, Tail dependence, Truncated vine},
	issn = {0047-259X},
	doi = {10.1016/j.jmva.2013.05.001},
	author = {Krupskii, Pavel and Joe, Harry}
}
@article { ISI:000320884400009,
	title = {Measures of tail asymmetry for bivariate copulas},
	journal = {Statistical Papers},
	volume = {54},
	number = {3},
	year = {2013},
	month = {AUG},
	pages = {709-726},
	publisher = {Springer},
	type = {Article},
	abstract = {Three tail asymmetry measures for bivariate copulas are introduced using two different approaches-univariate skewness of a projection and distance between a copula and its survival/reflected copula. We compare the asymmetry measures based on certain desirable properties. Bounds for each measure are obtained and also copulas which attain these extreme values are identified. Two data examples show the amount of asymmetry that might be expected in practice.},
	keywords = {Quantiles, Survival copula, Tail dependence, Univariate skewness},
	issn = {0932-5026},
	doi = {10.1007/s00362-012-0457-y},
	author = {Rosco, J. F. and Joe, Harry}
}
@article { ISI:000299151700002,
	title = {Tail risk of multivariate regular variation},
	journal = {Methodology and Computing in Applied Probability},
	volume = {13},
	number = {4},
	year = {2011},
	month = {DEC},
	pages = {671-693},
	publisher = {Springer},
	type = {Article},
	abstract = {Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes multivariate regular variation. Tractable bounds for tail risk are derived in terms of the tail dependence function that describes extremal dependence. Various examples involving Archimedean copulas are presented to illustrate the results and quality of the bounds.},
	keywords = {Coherent risk, Copula, Regularly varying, Tail conditional expectation, Tail dependence},
	issn = {1387-5841},
	doi = {10.1007/s11009-010-9183-x},
	author = {Joe, Harry and Li, Haijun}
}
