## Comparison of semiparametric and parametric methods for estimating copulas.(English)Zbl 1161.62364

Summary: Copulas have attracted significant attention in the recent literature for modeling multivariate observations. An important feature of copulas is that they enable us to specify the univariate marginal distributions and their joint behavior separately. The copula parameter captures the intrinsic dependence between the marginal variables and it can be estimated by parametric or semiparametric methods. For practical applications, the so called inference function for margins (IFM) method has emerged as the preferred fully parametric method because it is close to maximum likelihood (ML) in approach and is easier to implement. The purpose of this paper is to compare the ML and IFM methods with a semiparametric (SP) method that treats the univariate marginal distributions as unknown functions.
We consider the SP method proposed by C. Genest et al. [A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82, No. 3, 543–552 (1995; Zbl 0831.62030)], which has attracted considerable interest in the literature. The results of an extensive simulation study reported here show that the ML/IFM methods are nonrobust against misspecification of the marginal distributions, and that the SP method performs better than the ML and IFM methods, overall. A data example on household expenditure is used to illustrate the application of various data analytic methods for applying the SP method, and to compare and contrast the ML, IFM and SP methods. The main conclusion is that, in terms of statistical computations and data analysis, the SP method is better than ML and IFM methods when the marginal distributions are unknown which is almost always the case in practice.

### MSC:

 62H12 Estimation in multivariate analysis 62G05 Nonparametric estimation

Zbl 0831.62030

QRM
Full Text:

### References:

 [1] Bauwens, L.; Laurent, S., A new class of multivariate skew densities, with application to generalized autoregressive conditional heteroscedasticity models, J. business econom. statist., 23, 346-354, (2005) [2] Cherubini, U.; Luciano, E.; Vecchiato, W., Copula methods in finance, (2004), Wiley Chichester, UK · Zbl 1163.62081 [3] Genest, C.; MacKay, J., The joy of copulas: bivariate distributions with uniform marginals, Amer. statist., 40, 280-283, (1986) [4] Genest, C.; Rivest, L.-P., Statistical inference procedures for bivariate Archemedean copulas, J. amer. statist. assoc., 88, 1034-1043, (1993) · Zbl 0785.62032 [5] Genest, C.; Werker, B.J.M., Conditions on the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models, (), 103-112 · Zbl 1142.62330 [6] Genest, C.; Ghoudi, K.; Rivest, L.-P., A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika, 82, 3, 543-552, (1995) · Zbl 0831.62030 [7] Genest, C.; Quessy, J.-F.; Rémillard, B., Goodness-of-fit procedures for copula models based on the probability integral transformation, Scand. J. statist., 33, 337-366, (2006) · Zbl 1124.62028 [8] Granger, C.W., Terasvirta, T., Patton, A.J., 2005. Common factors in conditional distributions for bivariate time series. J. Econometrics, in press. · Zbl 1337.62263 [9] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517 [10] Joe, H., Asymptotic efficiency of the two-stage estimation method for copula-based models, J. multivariate anal., 94, 401-419, (2005) · Zbl 1066.62061 [11] McNeil, A.J.; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques, and tools, (2005), Wiley NY · Zbl 1089.91037 [12] Nelsen, R.B., An introduction to copulas, (2006), Springer New York · Zbl 1152.62030 [13] Patton, A.J., On the out-of-sample importance of skewness and asymmetric dependence for asset allocation, J. finan. econometrics, 2, 1, 130-168, (2004) [14] Patton, A.J., Estimation of multivariate models for time series of possibly different lengths, J. appl. econometrics, 21, 147-173, (2005) [15] Patton, A.J., Modelling asymmetric exchange rate dependence, Internat. econom. rev., 47, 527-556, (2005) [16] Shih, J.; Louis, T., Inferences on the association parameter in copula models for bivariate survival data, Biometrics, 51, 1384-1399, (1995) · Zbl 0869.62083 [17] Sklar, A., Fonctions de répartition à n dimensionset leurs marges, Publ. inst., statist. univ. Paris, 8, 229-231, (1959) [18] Tsukahara, H., Semiparametric estimation in copula models, Canad. J. statist., 33, 357-375, (2005) · Zbl 1077.62022 [19] Wang, W.; Ding, A.A., On assessing the association for bivariate current status data, Biometrika, 87, 4, 879-893, (2000) · Zbl 1028.62077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.