Maximum pseudo‐likelihood estimation based on estimated residuals in copula semiparametric models

Abstract

This paper deals with an estimation of the dependence structure of a multidimensional response variable in the presence of a multivariate covariate. It is assumed that the covariate affects only the marginal distributions through regression models while the dependence structure, which is described by a copula, is unaffected.A parametric estimation of the copula function is considered with focus on the maximum pseudo-likelihood method. It is proved that under some appropriate regularity assumptions the estimator calculated from the residuals has the same asymptotic distribution as the estimator based on the unobserved errors. In such case one can ignore the fact that the response is first adjusted for the effect of the covariate. The theoretical results are accompanied by a Monte Carlo simulation study which illustrates that the maximum pseudo-likelihood estimator based on residuals may behave poorly when the stated regularity assumptions are not satisfied.

Estimation and Inference in Factor Copula Models with Exogenous Covariates

“…D2) are similar to regularity conditions imposed byNeumeyer et al (2019, p. 141); as discussed inCôté et al (2019) andOmelka et al (2020), this assumption can be relaxed at the expense of additional technicalities. Assumption (D3) summarizes conditions which, in conjunction with the remaining assumptions, ensure the asymptotic equicontinuity of θ q → BT,S,i,j (θ q ).…”

Parametric copula adjusted for non- and semiparametric regression

“…( 17)) and Spearman's rho achieve the same asymptotic distribution as their oracle counterparts, even when the convergence rate of the estimator B to B is almost as slow as n −1/4 , and provided explicit rates of convergence. Omelka, Hudecová and Neumeyer (2020) studied model (1), but with parametrically specified m k and σ k . See also Veraverbeke, Omelka and Gijbels (2011); Veraverbeke, Gijbels and Omelka (2014); Gijbels, Omelka and Veraverbeke (2015) for related studies on empirical copula process involving a covariate.…”