Information bounds for Gaussian copula parameter in stationary semiparametric Markov models

Let ${\left\{V_t\right\}}_{t=1}^n$ be any univariate stationary first-order semiparametric Markov process generated from an unknown invariant marginal distribution and a bivariate Gaussian copula with unknown correlation coefficient ${\alpha }_0\in (-1,1)$. We prove that $\left(1-{\alpha }_0^2\right)$ is the semiparametric efficient variance bound for estimating the correlation parameter ${\alpha }_0$ in any Gaussian copula generated first-order stationary Markov models. Surprisingly, this variance bound is strictly larger than ${\left(1-{\alpha }_0^2\right)}^2$ (when ${\alpha }_0\ne 0$), which is the semiparametric efficient variance bound derived by Klaassen and Wellner (1997) for estimating the correlation parameter using any $i.i.d.$ data ${\left\{(X_i,Y_i)\right\}}_{i=1}^n$ generated from a bivariate Gaussian copula with two unknown marginal distributions.