Asymptotic efficiency of the two-stage estimation method for copula-based models. (English) Zbl 1066.62061

Summary: For multivariate copula-based models for which maximum likelihood is computationally difficult, a two-stage estimation procedure has been proposed previously; the first stage involves maximum likelihood from univariate margins, and the second stage involves maximum likelihood of the dependence parameters with the univariate parameters held fixed from the first stage. Using the theory of inference functions, a partitioned matrix in a form amenable to analysis is obtained for the asymptotic covariance matrix of the two-stage estimator. The asymptotic relative efficiency of the two-stage estimation procedure compared with maximum likelihood estimation is studied. Analysis of the limiting cases of the independence copula and Fréchet upper bound help to determine common patterns in the efficiency as the dependence in the model increases. For the Fréchet upper bound, the two-stage estimation procedure can sometimes be equivalent to maximum likelihood estimation for the univariate parameters. Numerical results are shown for some models, including multivariate ordinal probit and bivariate extreme value distributions, to indicate the typical level of asymptotic efficiency for discrete and continuous data.


62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
Full Text: DOI


[1] Aitchison, J.; Ho, C.H., The multivariate Poisson-log normal distribution, Biometrika, 76, 643-653, (1989) · Zbl 0679.62040
[2] Davis, P.J.; Rabinowitz, P., Methods of numerical integration, (1984), Academic Press Orlando · Zbl 0154.17802
[3] Frank, M.J., On the simultaneous associativity of \(F(x, y)\) and \(x + y + F(x, y)\), Aequationes math.,, 19, 194-226, (1979) · Zbl 0444.39003
[4] V.P. Godambe (Ed.), Estimating Functions, Oxford University Press, Oxford, 1991. · Zbl 0745.00006
[5] Gumbel, E.J., Distributions des valeurs extrêmes en plusieurs dimensions, Publ. inst. statist. univ. Paris, 9, 171-173, (1960) · Zbl 0093.15303
[6] Joe, H., Multivariate extreme value distributions with applications to environmental data, Canad. J. statist., 22, 47-64, (1994) · Zbl 0804.62052
[7] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[8] Jöreskog, K.G.; Moustaki, I., Factor analysis of ordinal variablesa comparison of three approaches, Multivariate behav. res., 36, 347-387, (2001)
[9] Lee, S.Y.; Poon, W.Y.; Bentler, P.M., A two-stage estimation of structural equation models with continuous and polytomous variables, Br. J. math. statist. psychol., 48, 339-358, (1995) · Zbl 0858.62100
[10] Heagerty, P.J.; Lele, S.R., A composite likelihood approach to binary spatial data, J. amer. statist. assoc., 93, 1099-1111, (1998) · Zbl 1064.62528
[11] Maydeu-Olivares, A., Multidimensional item response theory modeling of binary datalarge sample properties of NOHARM estimates, J. educ. behav. statist., 26, 49-69, (2001)
[12] Muthén, B., Contributions to factor analysis of dichotomous variables, Psychometrika, 43, 551-560, (1978) · Zbl 0394.62042
[13] Muthén, B., Latent variable structural equation modeling with categorical data, J. econom., 22, 43-65, (1979)
[14] Muthén, B., A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators, Psychometrika, 49, 115-132, (1984)
[15] Nash, J.C., Compact numerical methods for computerslinear algebra and function minimisation, (1990), Hilger New York
[16] Olsson, U., Maximum likelihood estimation of the polychoric correlation coefficient, Psychometrika, 44, 443-460, (1979) · Zbl 0428.62083
[17] Plackett, R.L., A class of bivariate distributions, J. amer. statist. assoc., 60, 516-522, (1965)
[18] Prescott, P.; Walden, A.T., Maximum likelihood estimation of the parameters of the generalized extreme-value distribution, Biometrika, 67, 723-724, (1980)
[19] Shih, J.H.; Louis, T.A., Inferences on the association parameter in copula models for bivariate survival data, Biometrics, 51, 1384-1399, (1995) · Zbl 0869.62083
[20] Smith, R.L., Maximum likelihood estimation in a class of nonregular cases, Biometrika, 72, 67-90, (1985) · Zbl 0583.62026
[21] Smith, R.L.; Tawn, J.A.; Yuen, H.-K., Statistics of multivariate extremes, Int. statist. inst. rev., 58, 47-58, (1990) · Zbl 0715.62095
[22] Tawn, J.A., Modelling multivariate extreme value distributions, Biometrika, 77, 245-253, (1990) · Zbl 0716.62051
[23] J.J. Xu, Statistical Modelling and Inference for Multivariate and Longitudinal Discrete Response Data. Ph.D. Thesis, Department of Statistics, University of British Columbia, 1996.
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