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The empirical beta copula. (English) Zbl 1360.62237

Summary: Given a sample from a continuous multivariate distribution \(F\), the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of \(F\). This idea can be regarded as a variant on R. Baker’s [J. Multivariate Anal. 99, No. 10, 2312–2327 (2008; Zbl 1151.62045)] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in [P. Janssen et al., J. Stat. Plann. Inference 142, No. 5, 1189–1197 (2012; Zbl 1236.62027)]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
62H05 Characterization and structure theory for multivariate probability distributions; copulas

Software:

R; copula; copula
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References:

[1] Baker, R., An order-statistics-based method for constructing multivariate distributions with fixed marginals, J. Multivariate Anal., 99, 2312-2327 (2008) · Zbl 1151.62045
[2] Bücher, A.; Dette, H., A note on bootstrap approximations for the empirical copula process, Statist. Probab. Lett., 80, 1925-1932 (2010) · Zbl 1202.62055
[3] Bücher, A.; Volgushev, S., Empirical and sequential empirical copula processes under serial dependence, J. Multivariate Anal., 119, 61-70 (2013) · Zbl 1277.62223
[4] Carley, H.; Taylor, M. D., A new proof of Sklar’s theorem, (Cuadras, C. M.; Fortiana, J.; Rodríguez-Lallena, J. A., Distributions with Given Marginals and Statistical Modelling (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 29-34 · Zbl 1137.62336
[5] Deheuvels, P., La fonction de dépendence empirique et ses propriétés, Un test non paramétrique d’indépendance, Bull. Cl. Sci. Acad. R. Belg. Sér. 5, 65, 274-292 (1979) · Zbl 0422.62037
[6] DeVore, R. A.; Lorentz, G. G., Constructive Approximations (1993), Springer: Springer Berlin-Heidelberg · Zbl 0797.41016
[7] Fermanian, J.-D.; Radulović, D.; Wegkamp, M. J., Weak convergence of empirical copula processes, Bernoulli, 10, 847-860 (2004) · Zbl 1068.62059
[8] Genest, C.; Nešlehová, J., A primer on copulas for count data, ASTIN Bull., 37, 475-515 (2007) · Zbl 1274.62398
[9] Genest, C.; Nešlehová, J. G.; Rémillard, B., On the empirical multilinear copula process for count data, Bernoulli, 20, 1344-1371 (2014) · Zbl 1365.62221
[11] Janssen, P.; Swanepoel, J.; Veraverbeke, N., Large sample behavior of the Bernstein copula estimator, J. Statist. Plann. Inference, 142, 1189-1197 (2012) · Zbl 1236.62027
[12] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman and Hall: Chapman and Hall London · Zbl 0990.62517
[13] Li, X.; Mikusiński, P.; Sherwood, H.; Taylor, M. D., On approximation of copulas, (Beneš, V.; Štěpán, J., Distributions with Given Marginals and Moment Problems (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 107-116 · Zbl 0905.60015
[14] Lorentz, G. G., Bernstein Polynomials (1986), Chelsea Publishing Company: Chelsea Publishing Company New York · Zbl 0989.41504
[15] Mai, J.-F.; Scherer, M., Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications (2012), Imperial College Press: Imperial College Press London
[16] Nelsen, R. B., An Introduction to Copulas (2006), Springer-Verlag: Springer-Verlag New York · Zbl 1152.62030
[18] Sancetta, A.; Satchell, S., The Bernstein copula and its applications to modeling and approximations of multivariate distributions, Econometric Theory, 20, 535-562 (2004) · Zbl 1061.62080
[19] Segers, J., Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions, Bernoulli, 18, 764-782 (2012) · Zbl 1243.62066
[20] Sklar, M., Fonctions de répartition á n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202
[21] Tsukahara, H., Semiparametric estimation in copula models, Canad. J. Statist., 33, 357-375 (2005), [Erratum: Canadian Journal of Statistics, 39, 734-735 (2011)] · Zbl 1077.62022
[22] Van der Vaart, A. W.; Wellner, J. A., Weak Convergence and Empirical Processes: With Applications to Statistics (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0862.60002
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