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Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points

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Abstract

The main purpose of the present paper is to establish the asymptotic properties of pseudo maximum likelihood estimators of the parameters of a multiple change-point model in the multivariate copula models when marginal distributions are unspecified but the copula function is parametrized. A pseudo likelihood ratio-type statistic is proposed for testing a sequence of observations for no change in the copula parameter against possible changes. Finally, a weighted bootstrap procedure that aims at evaluating the limiting distributions is examined.

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References

  1. D. J. Aldous, Exchangeability and Related Topics, in Lecture Notes in Math., Vol. 1117:’ Ecole d”et’e de probabilit’es de Saint-Flour XIII-1983 (Springer, Berlin, 1985), pp. 1–198.

    Google Scholar 

  2. A. Aue and L. Horv’ath, “Structural Breaks in Time Series”, J. Time Ser. Anal. 34(1), 1–16 (2012).

    Article  MathSciNet  Google Scholar 

  3. P. Barbe and P. Bertail, The Weighted Bootstrap, in Lecture Notes in Statist. (Springer, New York, 1995), Vol. 98.

    Google Scholar 

  4. M. Basseville and I. V. Nikiforov, Detection of Abrupt Changes: Theory and Application, in Prentice Hall Information and System Sciences Series (Prentice Hall, Englewood Cliffs, NJ, 1993).

    Google Scholar 

  5. P. Billingsley, Convergence of ProbabilityMeasures (Wiley, New York, 1968).

    Google Scholar 

  6. S. Bouzebda, “On the Strong Approximation of Bootstrapped Empirical Copula Processes with Applications”, Math. Methods Statist. 21(3), 153–188 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Bouzebda and M. Cherfi, “Test of Symmetry Based on Copula Function”, J. Statist. Plann. Inference 142(5), 1262–1271 (2012a).

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Bouzebda and M. Cherfi, “General Bootstrap for Dual gf-Divergence Estimates”, J. Probab. Stat., 2012 Art.ID 834107, 33 pages, doi:10.1155/2012/834107 (2012b).

  9. S. Bouzebda and A. Keziou, “A Test of Independence in Some Copula Models”, Math. Methods Statist. 17(2), 123–137 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Bouzebda and A. Keziou, “Estimation and Tests of Independence in Copula Models via Divergences”, C. R. Math. Acad. Sci. Paris 347(11–12), 667–672 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Bouzebda and A. Keziou, “New Estimates and Tests of Independence in Semiparametric Copula Models”, Kybernetika (Prague) 46(1), 178–201 (2010a).

    MATH  MathSciNet  Google Scholar 

  12. S. Bouzebda and A. Keziou, “A New Test Procedure of Independence in Copula Models via χ 2-Divergence”, Comm. Statist. TheoryMethods 39(1–2), 1–20 (2010b).

    MATH  MathSciNet  Google Scholar 

  13. S. Bouzebda and A. Keziou, “A Semiparametric Test of Independence in Copula Models for Censored Data”, C. R. Math. Acad. Sci. Paris 348(7–8), 449–453 (2010c).

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Bouzebda and A. Keziou, “A Semiparametric Maximum Likelihood Ratio Test for the Change Point in Copula Models”, Statist. Methodol. 14, 39–61 (2013).

    Article  MathSciNet  Google Scholar 

  15. S. Bouzebda and N. Limnios, “On General Bootstrap of Empirical Estimator of a Semi-Markov Kernel with Applications”, J. Multivariate Anal. 116, 52–62 (2013b).

    Article  MATH  MathSciNet  Google Scholar 

  16. J. V. Braun and H.-G. Müller, “Statistical Methods for DNA Sequence Segmentation”, Statist. Sci. 13(2), 142–162 (1998).

    Article  MATH  Google Scholar 

  17. B. E. Brodsky and B. S. Darkhovsky, Nonparametric Methods in Change-Point Problems, in Mathematics and Its Applications (Kluwer, Dordrecht, 1993), Vol. 243.

    Google Scholar 

  18. J. Chan, L. Horv’ath, and M. Huškov’a, “Darling-Erdős Limit Results for Change-Point Detection in Panel Data”, J. Statist. Plann. Inference. 143(5), 955–970 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Cheng, “A Note on Bootstrap Moment Consistency for Semiparametric M-Estimation”, arXiv:1109.4204 (2011).

    Google Scholar 

  20. J. Chen and A. K. Gupta, Parametric Statistical Change Point Analysis (Birkhäuser, Boston, MA, 2000).

    Book  MATH  Google Scholar 

  21. G. Cheng and J. Z. Huang, “Bootstrap Consistency for General Semiparametric M-Estimation”, Ann. Statist. 38(5), 2884–2915 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  22. X. Chen and Y. Fan, “Pseudo-Likelihood Ratio Tests for Semiparametric Multivariate Copula Model Selection”, Canad. J. Statist. 33(3), 389–414 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  23. X. Chen and Y. Fan, “Estimation of Copula-Based Semiparametric Time Series Models”, J. Econometrics 130(2), 307–335 (2006).

    Article  MathSciNet  Google Scholar 

  24. X. Chen, Y. Fan, D. Pouzo, and Z. Ying, “Estimation and Model Selection of Semiparametric Multivariate Survival Functions under General Censorship”, J. Econometrics 157(1), 129–142 (2010).

    Article  MathSciNet  Google Scholar 

  25. U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance, in Wiley Finance Series (Wiley, Chichester, 2004).

    Google Scholar 

  26. M. Csörgő and L. Horv’ath, “Invariance Principles for Change Point Problems”, J. Multivariate Anal. 27(1), 151–168 (1988).

    Article  MathSciNet  Google Scholar 

  27. M. Csörgő and L. Horváth, Nonparametric Methods for Change Point Problems, in Handbook of Statistics (Elsevier, North-Holland, Amsterdam, 1988), Vol. 7, pp. 403–425.

    Google Scholar 

  28. M. Csörgő and L. Horv’ath, Limit Theorems in Change-Point Analysis, in Wiley Series in Probab. and Statist. (Wiley, Chichester, 1997).

    Google Scholar 

  29. S. Cui and Y. Sun, “Checking for the Gamma Frailty Distribution under the Marginal Proportional Hazards Frailty Model”, Statist. Sinica 14(1), 249–267 (2004).

    MATH  MathSciNet  Google Scholar 

  30. P. Deheuvels, “La fonction de d’ependance empirique et ses propri’et’es. Un test non param’etrique d’ind’ependance””, Acad. Roy. Belg. Bull.Cl. Sci. (5) 65(6), 274–292 (1979).

    MATH  MathSciNet  Google Scholar 

  31. P. Deheuvels, “A Multivariate Bahadur-Kiefer Representation for the Empirical Copula Process”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI) 364, 120–147 (2009).

    MathSciNet  Google Scholar 

  32. A. Dias and P. Embrechts, “Change-Point Analysis for Dependence Structures in Finance and Insurance”, in Novos Rumos em Estat’istica, Ed. by C. Carvalho, F. Brilhante, and F. Rosado (Sociedade Portuguesa de Estat’istica, Lisbon, 2002), pp. 69–86.

    Google Scholar 

  33. M. Döring, “Multiple Change-Point Estimation with U-Statistics”, J. Statist. Plann. Inference 140(7), 2003–2017 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  34. A. Dvoretzky, J. Kiefer, and J. Wolfowitz, “Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator”, Ann. Math. Statist. 27, 642–669 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  35. B. Efron, “Bootstrap Methods: Another Look at the Jackknife”, Ann. Statist. 7(1), 1–26 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  36. C. M. Erasmus and F. Lombard, “Asymptotic Distributions of Quadratic Forms Occurring in Change-Point Problems”, Canad. J. Statist. 16(3), 259–268 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  37. D. Ferger, “On the Power of Nonparametric Changepoint-Tests”, Metrika 41(5), 277–292 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  38. J. P. Fine, J. Yan, and M. R. Kosorok, “Temporal Process Regression”, Biometrika 91(3), 683–703 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  39. E.W. Frees and E. A. Valdez, “Understanding Relationships Using Copulas”, North Amer.Actuar. J. 2(1), 1–25 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  40. Y.-X. Fu and R. N. Curnow, “Maximum Likelihood Estimation of Multiple Change Points”, Biometrika 77(3), 563–573 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  41. L. A. Gardner, Jr., “On Detecting Changes in the Mean of Normal Variates”, Ann. Math. Statist. 40, 116–126 (1969).

    Article  MathSciNet  Google Scholar 

  42. C. Genest, K. Ghoudi, and L.-P. Rivest, “A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distributions”, Biometrika 82(3), 543–552 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  43. E. Gombay, “The Weighted Sequential Likelihood Ratio”, Canad. J. Statist. 24(2), 229–239 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  44. E. Gombay, “The Likelihood Ratio under Noncontiguous Alternatives”, Canad. J. Statist. 25(3), 417–423 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  45. E. Gombay and L. Horv’ath, “An Application of the Maximum Likelihood Test to the Change-Point Problem”, Stochastic Process. Appl. 50(1), 161–171 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  46. D. Gu’egan and J. Zhang, “Change Analysis of a Dynamic Copula for Measuring Dependence in Multivariate Financial Data”, Quant. Finance, 10(4), 421–430 (2010).

    Article  MathSciNet  Google Scholar 

  47. H. He and T. A. Severini, “Asymptotic Properties of Maximum Likelihood Estimators in Models with Multiple Change Points”, Bernoulli 16(3), 759–779 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  48. L. Horv’ath and M. Huškov’a, “Testing for Changes Using Permutations of U-Statistics”, J. Statist. Plann. Inference 128(2), 351–371 (2005).

    Article  MathSciNet  Google Scholar 

  49. L. Horv’ath and M. Huškov’a, “Change-Point Detection in Panel Data”, J. Time SeriesAnal. 33(4), 631–648 (2012).

    Article  Google Scholar 

  50. H. Joe, “Parametric Families of Multivariate Distributions with Given Margins”, J. Multivariate Anal. 46(2), 262–282 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  51. H. Joe, Multivariate Models and Dependence Concepts, in Monographs on Statistics and Applied Probability (Chapman & Hall, London, 1997), Vol. 73.

    Google Scholar 

  52. Copula Theory and Its Applications, Ed. by P. Jaworski, F. Durante, W. Härdle, and T. Rychlik, in Lecture Notes in Statistics-Proceedings (Springer, Heidelberg, 2010), Vol 198.

    MATH  Google Scholar 

  53. O. Kallenberg, Foundations of Modern Probability, in Probability and its Applications (New York) (Springer, New York, 2002), 2nd ed.

    Google Scholar 

  54. G. Kimeldorf and A. Sampson, “One-Parameter Families of Bivariate Distributions with Fixed Marginals”, Comm. Statist. 4, 293–301 (1975a).

    Article  MATH  MathSciNet  Google Scholar 

  55. G. Kimeldorf and A. Sampson, “Uniform Representations of Bivariate Distributions”, Comm. Statist. 4(7), 617–627 (1975b).

    Article  MATH  MathSciNet  Google Scholar 

  56. I. Kojadinovic, J. Yan, and M. Holmes, “Fast Large-Sample Goodness-of-Fit Tests for Copulas”, Statist. Sinica 21, 841–871 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  57. M. R. Kosorok, Introduction to Empirical Processes and Semiparametric Inference, in Springer Series in Statist. (Springer, New York, 2008).

    Google Scholar 

  58. K.-Y. Liang and S. G. Self, “On the Asymptotic Behaviour of the Pseudolikelihood Ratio Test Statistic”, J. Roy. Statist. Soc. Ser. B 58(4), 785–796 (1996).

    MATH  MathSciNet  Google Scholar 

  59. D. Y. Lin, T. R. Fleming, and L. J. Wei, “Confidence Bands for Survival Curves under the Proportional Hazards Model”, Biometrika 81(1), 73–81 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  60. A. Y. Lo, “A Bayesian Method for Weighted Sampling”, Ann. Statist. 21(4), 2138–2148 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  61. D. M. Mason and M. A. Newton, “A Rank Statistics Approach to the Consistency of a General Bootstrap”, Ann. Statist. 20(3), 1611–1624 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  62. A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management. Concepts, Techniques and Tools, in Princeton Series in Finance (Princeton Univ. Press, Princeton, NJ, 2005).

    Google Scholar 

  63. O. Na, J. Lee, and S. Lee, “Change Point Detection in Copula ARMA-GARCH Models”, J. Time Series Anal. 33(4), 554–569 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  64. R. B. Nelsen, An Introduction to Copulas. in Springer Series in Statist. (Springer, New York, 2006), 2nd ed.

    Google Scholar 

  65. D. Oakes, “Multivariate Survival Distributions”, J. Nonparam. Statist. 3(3–4), 343–354 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  66. M. Orasch, “Using U-Statistics Based Processes to Detect Multiple Change-Points”, in Fields Inst. Commun., Vol. 44: Asymptotic Methods in Stochastics (Amer. Math. Soc., Providence, RI, 2004), pp. 315–334.

    Google Scholar 

  67. J. Prætgaard and J. A. Wellner, “Exchangeably Weighted Bootstraps of the General Empirical Process”, Ann. Probab. 21(4), 2053–2086 (1993).

    Article  MathSciNet  Google Scholar 

  68. E. S. Page, “Continuous Inspection Schemes”, Biometrika 41, 100–115 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  69. E. S. Page, “A test for a Change in a Parameter Occurring at an Unknown Point”, Biometrika 42, 523–527 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  70. M. Pauly, “Consistency of the Subsample Bootstrap Empirical Process”, Statistics 46(5), 621–626 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  71. B. R’emillard and O. Scaillet, “Testing for Equality between Two Copulas”, J. Multivariate Anal. 100(3), 377–386 (2009).

    Article  MathSciNet  Google Scholar 

  72. L. Rüschendorf, “Asymptotic Distributions of Multivariate Rank Order Statistics”, Ann. Statist. 4(5), 912–923 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  73. L. Rüschendorf, “On the Distributional Transform, Sklar’s Theorem, and the Empirical Copula Process”, J. Statist. Plann. Inference 139(11), 3921–3927 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  74. F.H Ruymgaart, “Asymptotic Normality of Nonparametric Tests for Independence”, Ann. Statist. 2, 892–910 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  75. F. H. Ruymgaart, G. R. Shorack, and W. R. van Zwet, “Asymptotic Normality of Nonparametric Tests for Independence”, Ann. Math. Statist. 43, 1122–1135 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  76. O. Scaillet, “A Kolmogorov-Smirnov Type Test for Positive Quadrant Dependence”, Canad. J. Statist. 33(3), 415–427 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  77. M. J. Schervish, Theory of Statistics, in Springer Series in Statist. (Springer, New York, 1995).

    Google Scholar 

  78. 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(398), 605–610 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  79. J. Shao and D. S. Tu, The Jackknife and Bootstrap, in Springer Series in Statist. (Springer, New York, 1995).

    Google Scholar 

  80. J.H. Shih and T. A. Louis, “Inferences on the Association Parameter in Copula Models for Bivariate Survival Data”, Biometrics 51(4), 1384–1399 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  81. A. Sklar, Fonctions de r’epartition á n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959).

    MathSciNet  Google Scholar 

  82. H. Tsukahara, “Semiparametric Estimation in Copula Models”, Canad. J. Statist. 33(3), 357–375 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  83. L. Y. Vostrikova, “Detection of “Discordance” of a Wiener Process”, Teor. Veroyatnost. i Primenen. 26(2), 362–368 (1981).

    MATH  MathSciNet  Google Scholar 

  84. W. Wang and A. A. Ding, “On Assessing the Association for Bivariate Current Status Data”, Biometrika 87(4), 879–893 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  85. X. Wang, C. van Eeden, and J. V. Zidek, “A symptotic Properties of Maximum Weighted Likelihood Estimators”, J. Statist. Plann. Inference 119(1), 37–54 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  86. A. W. van der Vaart and J. A. Wellner,Weak Convergence and Empirical Processes, with Applications to Statistics, in Springer Series in Statist. (Springer, New York, 1996).

    Google Scholar 

  87. J. A. Wellner and Y. Zhan, Bootstrapping Z-Estimators, Techn. Report No. 308 (Univ. Washington, Seattle, July 1996).

    Google Scholar 

  88. C. F. J. Wu, On the Asymptotic Property of the Jackknife Histogram, Techn. report (Dept. of Statistics, Univ. of Wisconsin, Madison, 1987).

    Google Scholar 

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    S. Bouzebda

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Bouzebda, S. Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points. Math. Meth. Stat. 23, 38–65 (2014). https://doi.org/10.3103/S1066530714010037

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