A composite Bayesian approach for quantile curve fitting with non-crossing constraints

To fit a set of quantile curves, Bayesian simultaneous quantile curve fitting methods face some challenges in properly specifying a feasible formulation and efficiently accommodating the non-crossing constraints. In this article, we propose a new minimization problem and develop its corresponding Bayesian analysis. The new minimization problem imposes two penalties to control not only the smoothness of fitted quantile curves but also the differences between quantile curves. This enables a direct inference on differences of quantile curves and facilitates improved information sharing among quantiles. After adopting B-spline approximation for the positive smoothing functions in the minimization problem, we specify the pseudo composite asymmetric Laplace likelihood and derive its priors. The computation algorithm, including partially collapsed Gibbs sampling for model parameters and Monte Carlo Expectation-Maximization algorithm for penalty parameters, are provided to carry out the proposed approach. The extensive simulation studies show that, compared with other candidate methods, the proposed approach yields more robust estimation. More advantages of the proposed approach are observed for the extreme quantiles, heavy-tailed random errors, and inference on the differences of quantiles. We also demonstrate the relative performances of the proposed approach and other competing methods through two real data analyses.