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Effective estimation algorithm for parameters of multivariate Farlie-Gumbel-Morgenstern copula. (English) Zbl 1478.62113

Summary: This paper focuses on the parameter estimation for the \(d\)-variate Farlie-Gumbel-Morgenstern (FGM) copula \((d\ge 2)\), which has \(2^d-d-1\) dependence parameters to be estimated; therefore, maximum likelihood estimation is not practical for a large \(d\) from the viewpoint of computational complexity. Besides, the restriction for the FGM copula’s parameters becomes increasingly complex as \(d\) becomes large, which makes parameter estimation difficult. We propose an effective estimation algorithm for the \(d\)-variate FGM copula by using the method of inference functions for margins under the restriction of the parameters. We then discuss its asymptotic normality as well as its performance determined through simulation studies. The proposed method is also applied to real data analysis of bearing reliability.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H12 Estimation in multivariate analysis
62E20 Asymptotic distribution theory in statistics
62N05 Reliability and life testing

Software:

CopulaModel
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Full Text: DOI

References:

[1] Cossette, H., Cote, M. P., Marceau, E., & Moutanabbir, K. (2013). Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation. Insurance: Mathematics and Economics, 52(3), 560-572. doi:10.1016/j.insmatheco.2013.03.006. · Zbl 1284.60027
[2] Cramer, H., Mathematical methods of statistics (1945), Princeton: Princeton University Press, Princeton · Zbl 0985.62001
[3] Eryilmaz, S.; Tank, F., On reliability analysis of a two-dependent-unit series system with a standby unit, Applied Mathematics and Computation, 218, 15, 7792-7797 (2012) · Zbl 1238.62122
[4] Farlie, DJ, The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47, 3-4, 307-323 (1960) · Zbl 0102.14903
[5] Gumbel, EJ, Bivariate exponential distributions, Journal of the American Statistical Association, 55, 292, 698-707 (1960) · Zbl 0099.14501
[6] Jaworski, P.; Durante, F.; Härdle, W.; Rychlik, T., Copula theory and its applications (2010), Berlin: Springer, Berlin
[7] Joe, H., & Xu, J. J. (1996). The estimation method of inference functions for margins for multivariate models. Technical Report, Department of Statistics, University of British Columbia, 166, 1-21. doi:10.14288/1.0225985.
[8] Joe, H., Asymptotic efficiency of the two-stage estimation method for copula-based models, Journal of Multivariate Analysis, 94, 2, 401-419 (2005) · Zbl 1066.62061
[9] Joe, H., Dependence Modeling with Copulas (2014), London: Chapman & Hall, London · Zbl 1346.62001
[10] Johnson, NL; Kotz, S., On some generalized Farlie-Gumbel-Morgenstern distributions, Communications in Statistics, 4, 5, 415-427 (1975) · Zbl 0342.62006
[11] Johnson, NL; Kotz, S., On some generalized Farlie-Gumbel-Morgenstern distributions-II regression, correlation and further generalizations, Communications in Statistics—Theory and Methods, 6, 6, 485-496 (1977) · Zbl 0382.62040
[12] Online document Lee, J., Qiu, H., Yu, G., Lin, J. & Rexnord Technical Services. (2007). Bearing Data Set. IMS, University of Cincinnati. NASA Ames Prognostics Data Repository. https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/. Accessed 1 2021.
[13] Lehmann, EL; Casella, G., Theory of Point Estimation (1998), New York: Springer, New York · Zbl 0916.62017
[14] MacDonald, IL, Does Newton-Raphson really fail?, Statistical Methods in Medical Research, 23, 3, 308-311 (2014)
[15] McCool, JI, Testing for dependency of failure times in life testing, Technometrics, 48, 1, 41-48 (2012)
[16] Morgenstern, D., Einfache Beispiele zweidimensionaler Verteilungen, Mitteilungsblatt für Mathematische Statistik, 8, 234-235 (1956) · Zbl 0070.36202
[17] Navarro, J.; Ruiz, JM; Sandoval, CJ, Properties of coherent systems with dependent components, Communications in Statistics—Theory and Methods, 36, 175-191 (2007) · Zbl 1121.60015
[18] Navarro, J.; Durante, F., Copula-based representations for the reliability of the residual lifetimes of coherent systems with dependent components, Journal of Multivariate Analysis, 158, 87-102 (2017) · Zbl 1397.62391
[19] Nelsen, RB, An introduction to copulas (2006), New York: Springer, New York · Zbl 1152.62030
[20] Patton, AJ, Estimation of multivariate models for time series of possibly different lengths, Journal of Applied Econometrics, 21, 2, 147-173 (2006)
[21] Qiu, H.; Lee, J.; Linb, J.; Yuc, G., Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics, Journal of Sound and Vibration, 289, 1066-1090 (2006)
[22] Shih, JH; Chang, YT; Konno, Y.; Emura, T., Estimation of a common mean vector in bivariate meta-analysis under the FGM copula, Statistics, 53, 3, 673-695 (2019) · Zbl 1418.62236
[23] Xu, J. J. (1996). Statistical modeling and inference for multivariate and longitudinal discrete response data. Ph.D. thesis, Department of Statistics, University of British Columbia.
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