Bayesian Inference on Monotone Regression Quantile: Coverage and Rate Acceleration

For a univariate monotone regression function, the location where a specific value is attained is called a regression quantile. We study the coverage of a Bayesian credible interval for a regression quantile in a nonparametric monotone regression model, assuming that the quantile is unique and the regression function has a positive derivative. We consider piecewise constant functions with equal intervals and put independent normal priors on the step heights. To comply with the monotonicity constraint for the regression function, we induce a “projection-posterior” by imposing the monotonicity constraint on samples from the posterior distribution of the step-heights. We demonstrate two different interesting phenomena in this context. First, we show that the asymptotic coverage of a credible interval is higher than the credibility, the opposite of a phenomenon observed for credible regions for smooth functions. Further, targeted asymptotic coverage may be obtained using an appropriate lower credibility level. Next, we show that the posterior contraction rate for the regression quantile can be improved from n-1/3 n - 1 / 3 to n-1/2 n - 1 / 2 by sampling in two stages, sparing a fraction of the sampling budget to sample later from a credible interval obtained in the first stage. We also show that the coverage of a second-stage credible interval agrees with its credibility. We study the finite sample performance of the methods through a simulation study.