| Title:
|
New estimates and tests of independence in semiparametric copula models (English) |
| Author:
|
Bouzebda, Salim |
| Author:
|
Keziou, Amor |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
178-201 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce new estimates and tests of independence in copula models with unknown margins using $\phi$-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an interior or a boundary value of the parameter space. Simulation results show that the choice of $\chi^2$-divergence has good properties in terms of efficiency-robustness. (English) |
| Keyword:
|
dependence function |
| Keyword:
|
multivariate rank statistics |
| Keyword:
|
semiparametric inference |
| Keyword:
|
copulas |
| Keyword:
|
boundary |
| Keyword:
|
divergences |
| Keyword:
|
duality |
| MSC:
|
62F03 |
| MSC:
|
62F10 |
| MSC:
|
62F12 |
| MSC:
|
62G05 |
| MSC:
|
62G10 |
| MSC:
|
62H05 |
| MSC:
|
62H12 |
| MSC:
|
62H15 |
| idZBL:
|
Zbl 1187.62067 |
| idMR:
|
MR2666901 |
| . |
| Date available:
|
2010-06-02T19:52:45Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140046 |
| . |
| Reference:
|
[1] M. Broniatowski and A. Keziou: Minimization of $\phi $-divergences on sets of signed measures.Studia Sci. Math. Hungar. 43 (2006), 4, 403–442. MR 2273419 |
| Reference:
|
[2] S. Bouzebda and A. Keziou: A test of independence in some copula models.Math. Methods Statist. 17 (2008), 2, 123–137. MR 2429124 |
| Reference:
|
[3] M. Broniatowski and A. Keziou: Parametric estimation and tests through divergences and the duality technique.J. Multivariate Anal. 100 (2009), 1, 16–36. MR 2460474 |
| Reference:
|
[4] S. Bouzebda and S. Keziou: A new test procedure of independence in copula models via $\chi ^2$-divergence.Comm. Statist. Theory Methods 38 (2009), 20, to appear. MR 2654856 |
| Reference:
|
[5] N. Cressie and T. R. C. Read: Multinomial goodness-of-fit tests.J. Roy. Statist. Soc. Ser. B,46 (1984), 3, 440–464. MR 0790631 |
| Reference:
|
[6] P. Deheuvels: La fonction de dépendance empirique et ses propriétés.Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. (5), 65 (1979), 6, 274–292. Zbl 0422.62037, MR 0573609 |
| Reference:
|
[7] P. Deheuvels: Propriétés d’existence et propriétés topologiques des fonctions de dépendance avec applications à la convergence des types pour des lois multivariées.C. R. Acad. Sci. Paris Sér. A-B 288 (1979), 2, A145–A148. Zbl 0396.60019, MR 0524771 |
| Reference:
|
[8] P. Deheuvels: Nonparametric test of independence.In: Proc. Nonparametric Asymptotic Statistics, Rouen 1979 (Lecture Notes in Mathematics 821). Springer, Berlin 1980, pp. 95–107. MR 0604022 |
| Reference:
|
[9] P. Deheuvels: A Kolmogorov-Smirnov type test for independence and multivariate samples.Rev. Roumaine Math. Pures Appl. 26 (1981), 2, 213–226. Zbl 0477.62030, MR 0616038 |
| Reference:
|
[10] P. Deheuvels: A multivariate Bahadur–Kiefer representation for the empirical copula process.Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 364 (2009), 120–147. MR 2749128 |
| Reference:
|
[11] Z. Feng and C. E. McCulloch: Statistical inference using maximum likelihood estimation and the generalized likelihood ratio when the true parameter is on the boundary of the parameter space.Statist. Probab. Lett. 13 (1992), 4, 325–332. MR 1160755 |
| Reference:
|
[12] J.-D. Fermanian, D. Radulović, and M. Wegkamp: Weak convergence of empirical copula processes.Bernoulli 10 (2004), 5, 847–860. MR 2093613 |
| Reference:
|
[13] P. Gaenssler and W. Stute: Seminar on empirical processes.(Notes based on lectures given at the Seminar on Empirical Processes held at Schloß Mickeln, Düsseldorf, FRG, September 8–13, 1985). DMV Seminar, Bd. 9. Birkhäuser Verlag, Basel – Boston 1987. MR 0902803 |
| Reference:
|
[14] J. Galambos: Order statistics of samples from multivariate distributions.J. Amer. Statist. Assoc. 70 (1975) (351, part 1), 674–680. Zbl 0315.62022, MR 0405714 |
| Reference:
|
[15] C. Genest, K. Ghoudi, and L.-P. Rivest: A semiparametric estimation procedure of dependence parameters in multivariate families of distributions.Biometrika 82 (1995), 3, 543–552. MR 1366280 |
| Reference:
|
[16] E. J. Gumbel: Bivariate exponential distributions.J. Amer. Statist. Assoc. 55 (1960), 698–707. Zbl 0099.14501, MR 0116403 |
| Reference:
|
[17] J. Hüsler and R.-D. Reiss: Maxima of normal random vectors: between independence and complete dependence.Statist. Probab. Lett. 7 (1989), 4, 283–286. MR 0980699 |
| Reference:
|
[18] H. Joe: Parametric families of multivariate distributions with given margins.J. Multivariate Anal. 46 (1993), 2, 262–282. Zbl 0778.62045, MR 1240425 |
| Reference:
|
[19] H. Joe: Multivariate Models and Dependence Concepts ( Monographs on Statistics and Applied Probability, vol.73). Chapman & Hall, London 1997. MR 1462613 |
| Reference:
|
[20] A. Keziou: Dual representation of $\phi $-divergences and applications.C. R. Math. Acad. Sci. Paris 336 (2003), 10, 857–862. Zbl 1043.62002, MR 1990028 |
| Reference:
|
[21] A. Keziou and S. Leoni-Aubin: On empirical likelihood for semiparametric two-sample density ratio models.J. Statist. Plann. Inference 138 (2008), 4, 915–928. MR 2384498 |
| Reference:
|
[22] G. Kimeldorf and A. Sampson: One-parameter families of bivariate distributions with fixed marginals.Comm. Statist. 4 (1975), 293–301. MR 0370861 |
| Reference:
|
[23] G. Kimeldorf and A. Sampson: Uniform representations of bivariate distributions.Comm. Statist. 4 (1975), 7, 617–627. MR 0397989 |
| Reference:
|
[24] K.-Y. Liang and S. G. Self: On the asymptotic behaviour of the pseudolikelihood ratio test statistic.J. Roy. Statist. Soc. Ser. B 58 (1996), 4, 785–796. MR 1410191 |
| Reference:
|
[25] F. Liese and I. Vajda: Convex Statistical Distances.Teubner, Leipzig 1987. MR 0926905 |
| Reference:
|
[26] F. Liese and I. Vajda: On divergences and informations in statistics and information theory.IEEE Trans. Inform. Theory 52 (2006), 10, 4394–4412. MR 2300826 |
| Reference:
|
[27] B. V. M. Mendes, E. F. L. de Melo, and R. B. Nelsen: Robust fits for copula models.Comm. Statist. Simulation Comput. 36 (2007), 4–6, 997–1017. MR 2415700 |
| Reference:
|
[28] D. S. Moore and M. C. Spruill: Unified large-sample theory of general chi-squared statistics for tests of fit.Ann. Statist. 3 (1975), 599–616. MR 0375569 |
| Reference:
|
[29] R. B. Nelsen: An Introduction to Copulas.(Lecture Notes in Statistics, vol. 139.) Springer-Verlag, New York 1999. Zbl 1152.62030, MR 1653203 |
| Reference:
|
[30] D. Oakes: Multivariate survival distributions.J. Nonparametr. Statist. 3 (1994), 3–4, 343–354. MR 1291555 |
| Reference:
|
[31] R. T. Rockafellar: Convex Analysis.Princeton University Press, Princeton, N.J. 1970. MR 0274683 |
| Reference:
|
[32] L. Rüschendorf: Asymptotic distributions of multivariate rank order statistics.Ann. Statist. 4 (1976), 5, 912–923. MR 0420794 |
| Reference:
|
[33] L. Rüschendorf: On the distributional transform, Sklar’s theorem, and the empirical copula process.J. Statist. Plann. Inference 139 (2009), 11, 3921–3927. MR 2553778 |
| Reference:
|
[34] F. H. Ruymgaart: Asymptotic normality of nonparametric tests for independence.Ann. Statist. 2 (1974), 892–910. Zbl 0303.62040, MR 0386140 |
| Reference:
|
[35] F. H. Ruymgaart, G. R. Shorack, and W. R. van Zwet: Asymptotic normality of nonparametric tests for independence.Ann. Math. Statist. 43 (1972), 1122–1135. MR 0339397 |
| Reference:
|
[36] B. Schweizer: Thirty years of copulas.In: Advances in probability distributions with given marginals, Rome 1990, vol. 67, Math. Appl., pp. 13–50. Kluwer Academic Publishers, Dordrecht 1991. Zbl 0727.60001, MR 1215944 |
| Reference:
|
[37] S. G. Self and K.-Y. Liang: Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions.J. Amer. Statist. Assoc. 82 (1987), 398, 605–610. MR 0898365 |
| Reference:
|
[38] J. H. Shih and T. A. Louis: Inferences on the association parameter in copula models for bivariate survival data.Biometrics 51 (1995), 4, 1384–1399. MR 1381050 |
| Reference:
|
[39] A. Sklar: Random variables, joint distribution functions, and copulas.Kybernetika 9 (1973), 449–460. Zbl 0292.60036, MR 0345164 |
| Reference:
|
[40] M. Sklar: Fonctions de répartition à $n$ dimensions et leurs marges.Publ. Inst. Statist. Univ. Paris 8 (1959), 229–231. MR 0125600 |
| Reference:
|
[41] W. Stute: The oscillation behavior of empirical processes: The multivariate case.Ann. Probab. 12 (1984), 361–379. Zbl 0533.62037, MR 0735843 |
| Reference:
|
[42] H. Tsukahara: Semiparametric estimation in copula models.Canad. J. Statist. 33 (2005), 3, 357–375. Zbl 1077.62022, MR 2193980 |
| Reference:
|
[43] W. Wang and A. A. Ding: On assessing the association for bivariate current status data.Biometrika 87 (2000), 4, 879–893. MR 1813981 |
| . |
