Efficient Estimation of Semiparametric Multivariate Copula Models Efficient Estimation of Semiparametric Multivariate Copula Models *

We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the “curse of dimensionality” associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve further efficiency gains. Two such cases are studied: (a) the marginal distributions are equal but otherwise unspecified, and (b) some but not all marginal distributions are parametric. Monte Carlo studies indicate that the sieve MLEs perform well in finite samples, especially when prior information on the marginal distributions is incorporated.

[1]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[2]  D. Clayton.  A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence , 1978 .

[3]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[4]  S. Geman,et al.  Nonparametric Maximum Likelihood Estimation by the Method of Sieves , 1982 .

[5]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[6]  D. Oakes.  A Model for Association in Bivariate Survival Data , 1982 .

[7]  Lung-fei Lee.  Some Approaches to the Correction of Selectivity Bias , 1982 .

[8]  Lung-fei Lee.  Generalized Econometric Models with Selectivity , 1983 .

[9]  D. Clayton,et al.  Multivariate generalizations of the proportional hazards model , 1985 .

[10]  D. Oakes,et al.  Semiparametric inference in a model for association in bivanate survival data , 1986 .

[11]  C. Genest,et al.  The Joy of Copulas: Bivariate Distributions with Uniform Marginals , 1986 .

[12]  C. Genest.  Frank's family of bivariate distributions , 1987 .

[13]  A. Gallant,et al.  Semi-nonparametric Maximum Likelihood Estimation , 1987 .

[14]  P. Robinson.  ROOT-N-CONSISTENT SEMIPARAMETRIC REGRESSION , 1988 .

[15]  D. Oakes,et al.  Bivariate survival models induced by frailties , 1989 .

[16]  James J. Heckman,et al.  The identifiability of the competing risks model , 1989 .

[17]  C. J. Stone,et al.  Large-Sample Inference for Log-Spline Models , 1990 .

[18]  A. Barron,et al.  APPROXIMATION OF DENSITY FUNCTIONS BY SEQUENCES OF EXPONENTIAL FAMILIES , 1991 .

[19]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[20]  G. Maguluri.  Semiparametric Estimation of Association in a Bivariate Survival Function , 1993 .

[21]  D. Oakes.  Multivariate survival distributions , 1994 .

[22]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[23]  W. Wong,et al.  Convergence Rate of Sieve Estimates , 1994 .

[24]  W. Wong,et al.  Probability inequalities for likelihood ratios and convergence rates of sieve MLEs , 1995 .

[25]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[26]  C. Genest,et al.  A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , 1995 .

[27]  T. Louis,et al.  Inferences on the association parameter in copula models for bivariate survival data. , 1995, Biometrics.

[28]  Xiaotong Shen,et al.  On methods of sieves and penalization , 1997 .

[29]  C. Klaassen,et al.  Efficient estimation in the bivariate normal copula model: normal margins are least favourable , 1997 .

[30]  W. Newey,et al.  Convergence rates and asymptotic normality for series estimators , 1997 .

[31]  Emiliano A. Valdez,et al.  Understanding Relationships Using Copulas , 1998 .

[32]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[33]  Halbert White,et al.  Improved Rates and Asymptotic Normality for Nonparametric Neural Network Estimators , 1999, IEEE Trans. Inf. Theory.

[34]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[35]  Christian Genest,et al.  Conditions for the Asymptotic Semiparametric Efficiency of an Omnibus Estimator of Dependence Parame , 2000 .

[36]  Cross Validated Snp Density Estimates , 2000 .

[37]  S. Kotz,et al.  Correlation and dependence , 2001 .

[38]  S. R. Jammalamadaka,et al.  Empirical Processes in M-Estimation , 2001 .

[39]  Xiaotong Shen,et al.  Adaptive Model Selection , 2002 .

[40]  Common Factors in Conditional Distributions , 2002 .

[41]  Andrew J. Patton.  On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation , 2002 .

[42]  Andrew J. Patton,et al.  Common Factors in Conditional Distributions for Bivariate Time Series , 2003 .

[43]  Xiaohong Chen,et al.  Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , 2003 .

[44]  Xiaohong Chen,et al.  Estimation of Copula-Based Semiparametric Time Series Models , 2006 .

[45]  Xiaotong Shen,et al.  Inference After Model Selection , 2004 .

[46]  Xiaohong Chen,et al.  Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification , 2006 .