Transform MCMC Schemes for Sampling Intractable Factor Copula Models

In financial risk management, modelling dependency within a random vector $$\mathcal{X}$$ X is crucial, a standard approach is the use of a copula model. Say the copula model can be sampled through realizations of $$\mathcal{Y}$$ Y having copula function C: had the marginals of $$\mathcal{Y}$$ Y been known, sampling $$\mathcal{X}^{(i)}$$ X ( i ) , the i-th component of $$\mathcal{X}$$ X , would directly follow by composing $$\mathcal{Y}^{(i)}$$ Y ( i ) with its cumulative distribution function (c.d.f.) and the inverse c.d.f. of $$\mathcal{X}^{(i)}$$ X ( i ) . In this work, the marginals of $$\mathcal{Y}$$ Y are not explicit, as in a factor copula model. We design an algorithm which samples $$\mathcal{X}$$ X through an empirical approximation of the c.d.f. of the $$\mathcal{Y}$$ Y -marginals. To be able to handle complex distributions for $$\mathcal{Y}$$ Y or rare-event computations, we allow Markov Chain Monte Carlo (MCMC) samplers. We establish convergence results whose rates depend on the tails of $$\mathcal{X}$$ X , $$\mathcal{Y}$$ Y and the Lyapunov function of the MCMC sampler. We present numerical experiments confirming the convergence rates and also revisit a real data analysis from financial risk management.