Efficient parameter estimation via Gaussian copulas for quantile regression with longitudinal data

Specifying a correlation matrix is challenging in quantile regression with longitudinal data. A naive method is simply to adopt an independence working model. However, the efficiency of parameter estimates may be lost. We propose constructing a working correlation matrix via Gaussian copula which can handle or incorporate general serial dependence. A suit of unbiased estimating functions can be obtained by assuming the Gaussian copula with different correlation matrices, and the empirical likelihood method can then combine these unbiased estimating functions. Furthermore, the induced smoothing approach is applied to the discontinuous estimating functions to reduce computation burdens. The asymptotic normality of the resulting estimators is established. Simulation studies indicate that the proposed method is superior to the alternative estimating functions especially when the working correlation matrix is misspecified. Finally, a real dataset from forced expiratory volume study is used to illustrate the proposed method.