Hierarchical copulas with Archimedean blocks and asymmetric between-block pairs. (English) Zbl 1510.62225
Summary: A new class of hierarchical copulas is introduced based on joint survival functions of multivariate exponential mixture distributions. The key element of this construction is the mixing random vector defined by convolutions associated to a Lévy subordinator, and leading to hierarchical copulas with Archimedean within-block copulas and asymmetric between-block pair-copulas. For the specific case of two-level trees, dependence properties of pairs are investigated, and a full estimation procedure is proposed for the tree structure and parameters of the hierarchical copulas. The efficiency of the procedure is illustrated through three simulation examples and a study with two real datasets.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
Keywords:
hierarchical copulas; asymmetric pair-copulas; Archimedean copulas; composite likelihood estimation; block-exchangeability; partitioning around mdoidsReferences:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Vol. 55 (1948), US Government Printing Office |
| [2] | Applebaum, D., Lévy Processes and Stochastic Calculus (2009), Cambridge University Press · Zbl 1200.60001 |
| [3] | Barbakh, W. A.; Wu, Y.; Fyfe, C., Review of clustering algorithms, (Non-Standard Parameter Adaptation for Exploratory Data Analysis (2009), Springer), 7-28 |
| [4] | Bedford, T.; Cooke, R. M., Vines: A new graphical model for dependent random variables, Ann. Statist., 30, 1031-1068 (2002) · Zbl 1101.62339 |
| [5] | Belzile, L. R.; Nešlehová, J. G., Extremal attractors of Liouville copulas, J. Multivariate Anal., 160, 68-92 (2017) · Zbl 1372.60071 |
| [6] | Bernard, E.; Naveau, P.; Vrac, M.; Mestre, O., Clustering of maxima: Spatial dependencies among heavy rainfall in France, J. Clim., 26, 20, 7929-7937 (2013) |
| [7] | Bernardoff, P., Marshall-Olkin Laplace transform copulas of multivariate gamma distributions, Comm. Statist. Theory Methods, 47, 3, 655-670 (2018) · Zbl 1388.62150 |
| [8] | Capéraà, P.; Fougères, A.-L.; Genest, C., Bivariate distributions with given extreme value attractor, J. Multivariate Anal., 72, 1, 30-49 (2000) · Zbl 0978.62043 |
| [9] | Charpentier, A.; Fougères, A.-L.; Genest, C.; Nešlehová, J., Multivariate Archimax copulas, J. Multivariate Anal., 126, 118-136 (2014) · Zbl 1349.62173 |
| [10] | Cossette, H.; Gadoury, S.-P.; Marceau, É.; Mtalai, I., Hierarchical Archimedean copulas through multivariate compound distributions, Insurance Math. Econom., 76, 1-13 (2017) · Zbl 1395.62112 |
| [11] | Cossette, H.; Gadoury, S.-P.; Marceau, E.; Robert, C. Y., Composite likelihood estimation method for hierarchical Archimedean copulas defined with multivariate compound distributions, J. Multivariate Anal., 172, 59-83 (2019) · Zbl 1419.62120 |
| [12] | Czado, C., (Analyzing Dependent Data with Vine Copulas. Analyzing Dependent Data with Vine Copulas, Lecture Notes in Statistics (2019), Springer) · Zbl 1425.62001 |
| [13] | Feller, W., Introduction to the Theory of Probability and its Applications, Vols. 2, II (1971), Wiley: Wiley New York · Zbl 0219.60003 |
| [14] | Genest, C.; Mackay, R., Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données, Canad. J. Statist., 14, 2, 145-159 (1986) · Zbl 0605.62049 |
| [15] | Genest, C.; Nešlehová, J. G., Assessing and modeling asymmetry in bivariate continuous data, (Copulae in Mathematical and Quantitative Finance (2013), Springer), 91-114 · Zbl 1273.62112 |
| [16] | Genest, C.; Nešlehová, J.; Quessy, J.-F., Tests of symmetry for bivariate copulas, Ann. Inst. Statist. Math., 64, 4, 811-834 (2012) · Zbl 1440.62182 |
| [17] | Górecki, J.; Hofert, M.; Holeňa, M., An approach to structure determination and estimation of hierarchical Archimedean Copulas and its application to Bayesian classification, J. Intell. Inf. Syst., 46, 1, 21-59 (2016) |
| [18] | Górecki, J.; Hofert, M.; Holeňa, M., On structure, family and parameter estimation of hierarchical Archimedean copulas, J. Stat. Comput. Simul., 87, 17, 3261-3324 (2017) · Zbl 07192120 |
| [19] | Górecki, J.; Hofert, M.; Okhrin, O., Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation (2020), arXiv preprint arXiv:2003.13301 |
| [20] | Górecki, J.; Holeňa, M., Structure determination and estimation of hierarchical Archimedean copulas based on Kendall correlation matrix, (International Workshop on New Frontiers in Mining Complex Patterns (2013), Springer), 132-147 |
| [21] | Henderson, R.; Shimakura, S., A serially correlated gamma frailty model for longitudinal count data, Biometrika, 90, 2, 355-366 (2003) · Zbl 1034.62115 |
| [22] | Hering, C.; Hofert, M.; Mai, J.-F.; Scherer, M., Constructing hierarchical Archimedean copulas with Lévy subordinators, J. Multivariate Anal., 101, 6, 1428-1433 (2010) · Zbl 1194.60017 |
| [23] | Hofert, M., Sampling Archimedean copulas, Comput. Statist. Data Anal., 52, 12, 5163-5174 (2008) · Zbl 1452.62070 |
| [24] | Hofert, M., Sampling Nested Archimedean Copulas with Applications to CDO Pricing (2010), Universität Ulm, (Ph.D. thesis) |
| [25] | Hofert, M., Efficiently sampling nested Archimedean copulas, Comput. Statist. Data Anal., 55, 1, 57-70 (2011) · Zbl 1247.62132 |
| [26] | Hofert, M., Sampling exponentially tilted stable distributions, ACM Trans. Modeling Comput. Simul. (TOMACS), 22, 1, 1-11 (2011) · Zbl 1386.65051 |
| [27] | Hofert, M.; Huser, R.; Prasad, A., Hierarchical Archimax copulas, J. Multivariate Anal., 167, 195-211 (2018) · Zbl 1418.62223 |
| [28] | Hofert, M.; Mächler, M.; McNeil, A. J., Likelihood inference for Archimedean copulas in high dimensions under known margins, J. Multivariate Anal., 110, 133-150 (2012) · Zbl 1244.62073 |
| [29] | Hua, L., A note on upper tail behavior of Liouville copulas, Risks, 4, 4, 40 (2016) |
| [30] | Joe, H., (Families of \(m\)-Variate Distributions with Given Margins and \(M ( m - 1 ) / 2\) Bivariate Dependence Parameters. Families of \(m\)-Variate Distributions with Given Margins and \(M ( m - 1 ) / 2\) Bivariate Dependence Parameters, Lecture Notes-Monograph Series (1996), JSTOR), 120-141 |
| [31] | Joe, H., Multivariate Models and Multivariate Dependence Concepts (1997), CRC Press · Zbl 0990.62517 |
| [32] | Joe, H., Dependence Modeling with Copulas (2014), Chapman and Hall/CRC · Zbl 1346.62001 |
| [33] | Joe, H.; Kurowicka, D., Dependence Modeling: Vine Copula Handbook (2011), World Scientific |
| [34] | Khoudraji, A., Contributions à l’étude des copules et à la modélisation de valeurs extrêmes bivarées (1995), Université Laval, (Ph.D. thesis) |
| [35] | Kimberling, C. H., A probabilistic interpretation of complete monotonicity, Aequationes Math., 10, 2, 152-164 (1974) · Zbl 0309.60012 |
| [36] | Liebscher, E., Construction of asymmetric multivariate copulas, J. Multivariate Anal., 99, 10, 2234-2250 (2008) · Zbl 1151.62043 |
| [37] | Lindsay, B. G., Composite likelihood methods, Contemp. Math., 80, 1, 221-239 (1988) · Zbl 0672.62069 |
| [38] | MacQueen, J., et al., Some methods for classification and analysis of multivariate observations, in: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Oakland, CA, USA, 1967, pp. 281-297. · Zbl 0214.46201 |
| [39] | Marshall, A. W.; Olkin, I., Families of multivariate distributions, J. Am. Stat. Assoc., 83, 403, 834-841 (1988) · Zbl 0683.62029 |
| [40] | McNeil, A. J., Sampling nested Archimedean copulas, J. Stat. Comput. Simul., 78, 6, 567-581 (2008) · Zbl 1221.00061 |
| [41] | McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management: Concepts, Techniques and Tools (2015), Princeton University Press · Zbl 1337.91003 |
| [42] | McNeil, A. J.; Nešlehová, J., From Archimedean to Liouville copulas, J. Multivariate Anal., 101, 8, 1772-1790 (2010) · Zbl 1190.62102 |
| [43] | Mesiar, R.; Jagr, V., D-Dimensional dependence functions and Archimax copulas, Fuzzy Sets and Systems, 228, 78-87 (2013) · Zbl 1284.62345 |
| [44] | Nelsen, R., An Introduction to Copulas (2006), Springer Science Business Media: Springer Science Business Media New York · Zbl 1152.62030 |
| [45] | Okhrin, O.; Okhrin, Y.; Schmid, W., On the structure and estimation of hierarchical Archimedean copulas, J. Econometrics, 173, 2, 189-204 (2013) · Zbl 1443.62137 |
| [46] | Parner, E., A composite likelihood approach to multivariate survival data, Scand. J. Stat., 28, 2, 295-302 (2001) · Zbl 0973.62088 |
| [47] | Perreault, S.; Duchesne, T.; Nešlehová, J. G., Detection of block-exchangeable structure in large-scale correlation matrices, J. Multivariate Anal., 169, 400-422 (2019) · Zbl 1411.62038 |
| [48] | Quessy, J.-F.; Bahraoui, T., Graphical and formal statistical tools for the symmetry of bivariate copulas, Canad. J. Statist., 41, 4, 637-656 (2013) · Zbl 1351.62117 |
| [49] | Rousseeuw, P. J.; Kaufman, L., Finding Groups in Data (1990), Wiley Online Library: Wiley Online Library Hoboken · Zbl 1345.62009 |
| [50] | Savu, C.; Trede, M., Hierarchies of Archimedean copulas, Quant. Finance, 10, 3, 295-304 (2010) · Zbl 1270.91086 |
| [51] | Schweizer, B., Thirty years of copulas, (Advances in Probability Distributions with Given Marginals (1991), Springer), 13-50 · Zbl 0727.60001 |
| [52] | Shi, P.; Feng, X.; Boucher, J.-P., Multilevel modeling of insurance claims using copulas, Ann. Appl. Stat., 10, 2, 834-863 (2016) · Zbl 1400.62238 |
| [53] | Sklar, M., Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202 |
| [54] | Varin, C.; Reid, N.; Firth, D., An overview of composite likelihood methods, Statist. Sinica, 21, 5-42 (2011) · Zbl 1534.62022 |
| [55] | Varin, C.; Vidoni, P., A note on composite likelihood inference and model selection, Biometrika, 92, 3, 519-528 (2005) · Zbl 1183.62037 |
| [56] | Zhao, Y.; Joe, H., Composite likelihood estimation in multivariate data analysis, Canad. J. Statist., 33, 3, 335-356 (2005) · Zbl 1077.62045 |
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