@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:000327908200010,
	title = {Strength of tail dependence based on conditional tail expectation},
	journal = {Journal of Multivariate Analysis},
	volume = {123},
	year = {2014},
	month = {JAN},
	pages = {143-159},
	publisher = {Elsevier Inc},
	type = {Article},
	abstract = {We use the conditional distribution and conditional expectation of one random variable given the other one being large to capture the strength of dependence in the tails of a bivariate random vector. We study the tail behavior of the boundary conditional cumulative distribution function (cdf) and two forms of conditional tail expectation (CTE) for various bivariate copula families. In general, for nonnegative dependence, there are three levels of strength of dependence in the tails according to the tail behavior of CTEs: asymptotically linear, sub-linear and constant. For each of these three levels, we investigate the tail behavior of CTEs for the marginal distributions belonging to maximum domain of attraction of Frechet and Gumbel, respectively, and for copula families with different tail behavior. (C) 2013 Elsevier Inc. All rights reserved.},
	keywords = {Boundary conditional distribution, Copula, intermediate tail dependence, Maximum domain of attraction, Stochastic increasing, Tail behavior, Tail order, Tail quadrant independence},
	issn = {0047-259X},
	doi = {10.1016/j.jmva.2013.09.001},
	author = {Hua, Lei and Joe, Harry}
}
