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Semiparametric estimation in copula models. (English) Zbl 1077.62022
Can. J. Stat. 33, No. 3, 357-375 (2005); erratum ibid. 39, No. 4, 734 (2011).
Summary: The author recalls the limiting behaviour of the empirical copula process and applies it to prove some asymptotic properties of a minimum distance estimator for a Euclidian parameter in a copula model. The estimator in question is semiparametric in that no knowledge of the marginal distributions is necessary. The author also proposes another semiparametric estimator which he calls “rank approximate Z-estimator” and whose asymptotic normality he derives. He further presents Monte Carlo results for the comparison of various estimators in four well-known bivariate copula models.

MSC:
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
65C05 Monte Carlo methods
62G30 Order statistics; empirical distribution functions
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[1] M.Abramowitz & I. A.Stegun, eds. (1965). Handbook of Mathematical Functions. Dover, New York.
[2] Beran, Handbook of Statistics, Volume 4: Nonparametric Methods pp 741– (1984) · Zbl 0547.62023
[3] Bickel, Efficient and Adaptive Estimation for Semiparametric Models (1993) · Zbl 0786.62001
[4] Cebrian, Testing for concordance ordering, Astin Bulletin 34 pp 151– (2004)
[5] Clayton, A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika 65 pp 141– (1978) · Zbl 0394.92021
[6] Dabrowska, Weak convergence of a product integral dependence measure, Scandinavian Journal of Statistics 23 pp 551– (1996) · Zbl 0907.62052
[7] Dabrowska, Rank estimates in a class of semiparametric two-sample models, Annals of the Institute of Statistical Mathematics 41 pp 63– (1989) · Zbl 0723.62023
[8] Deheuvels, La fonction de dépendance empirique et ses propriétés: un test non paramétrique d’indépendance, Académie royale de Belgique, Bulletin de la classe des sciences: 5e série 65 pp 274– (1979)
[9] Donoho, The ”automatic” robustness of minimum distance functionals, The Annals of Statistics 16 pp 552– (1988) · Zbl 0684.62030
[10] Drouet-Mari, Correlation and Dependence (2001)
[11] Dvoretzky, Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, The Annals of Mathematical Statistics 27 pp 642– (1956) · Zbl 0073.14603
[12] Einmahl, Multivariate Empirical Processes (1987)
[13] Embrechts, Risk Management: Value at Risk and Beyond pp 176– (2002)
[14] Fermanian, Weak convergence of empirical copula processes, Bernoulli 10 pp 847– (2004) · Zbl 1068.62059
[15] J.-D. Fermanian & O. Scaillet (2004). Some statistical pitfalls in copula modeling for financial applications. HEC Genève: http://www.hec.unige.ch/recherches_publications/cahiers/2004/2004.02.pdf
[16] Frees, Understanding relationships using copulas, North American Actuarial Journal 2 pp 1– (1998) · Zbl 1081.62564
[17] Gänssler, Seminar on Empirical Processes (1987)
[18] Genest, Frank’s family of bivariate distributions, Biometrika 74 pp 549– (1987) · Zbl 0635.62038
[19] Genest, A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika 82 pp 543– (1995) · Zbl 0831.62030
[20] Genest, Distributions with Given Marginals and Statistical Modelling pp 103– (2002)
[21] Ghoudi, Asymptotic Methods in Stochastics: Festschrift for Miklós Csörg( pp 381– (2004)
[22] Glidden, A two-stage estimator of the dependence parameter for the Clayton-Oakes model, Lifetime Data Analysis 6 pp 141– (2000) · Zbl 1054.62113
[23] Hougaard, Analysis of Multivariate Survival Data (2000) · Zbl 0962.62096
[24] Joe, Multivariate Models and Dependence Concepts (1997) · Zbl 0990.62517
[25] Kiefer, On large deviations of the empiric D. F. of vector chance variables and a law of iterated logarithm, Pacific Journal of Mathematics 11 pp 649– (1961) · Zbl 0119.34904
[26] Kiefer, Nonparametric Techniques in Statistical Inference pp 299– (1970)
[27] Klaassen, Efficient estimation in the bivariate normal copula model: normal margins are least-favorable, Bernoulli 3 pp 55– (1997) · Zbl 0877.62055
[28] Maguluri, Semiparametric estimation of association in a bivariate survival function, The Annals of Statistics 21 pp 1648– (1993) · Zbl 0792.62027
[29] Millar, Robust estimation via minimum distance methods, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 55 pp 73– (1981)
[30] Millar, École d’été de probabilités de Saint-Flour XI-1981 pp 75– (1983)
[31] Nelsen, An Introduction to Copulas (1999) · Zbl 0909.62052
[32] Neuhaus, On weak convergence of stochastic processes with multidimensional time parameter, The Annals of Mathematical Statistics 42 pp 1285– (1971) · Zbl 0222.60013
[33] Oakes, A model for association in bivariate survival data, Journal of the Royal Statistical Society Series B 44 pp 414– (1982) · Zbl 0503.62035
[34] Ruymgaart, Asymptotic Theory of Rank Tests for Independence (1978)
[35] Shih, Inferences on the association parameter in copula models for bivariate survival data, Biometrics 51 pp 1384– (1995) · Zbl 0869.62083
[36] Sklar, Fonctions de répartition à n dimensions et leurs marges, Publications de I’Institut de statistique de l’Université de Paris 8 pp 229– (1959)
[37] Stute, The oscillation behavior of empirical processes: the multivariate case, The Annals of Probability 12 pp 361– (1984) · Zbl 0533.62037
[38] Tsukahara, A rank estimator in the two-sample transformation model with randomly censored data, Annals of the Institute of Statistical Mathematics 44 pp 313– (1992) · Zbl 0763.62017
[39] H. Tsukahara (2000). Empirical Copulas and Some Applications. Research Report No 27, The Institute for Economic Studies, Seijo University: http://www.seijo.ac.jp/research/keiken/green/green27.pdf
[40] van der Vaart, Weak Convergence and Empirical Processes: With Applications to Statistics (1996) · Zbl 0862.60002
[41] Wang, On assessing the association for bivariate current status data, Biometrika 87 pp 879– (2000) · Zbl 1028.62077
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