Pyramid quantile regression

Noncrossing simultaneous Bayesian quantile curve fitting

“…Rodrigues and Fan (2017) proposed a procedure to postprocess crossing quantiles in a Bayesian framework, however its performance is still influenced by the initial estimates. Recently, several authors have argued that simultaneous estimation offers better estimates, better global efficiency for the estimators have been observed empirically in several studies (Reich and Smith 2013, Yang and Tokdar 2017, Fang et al 2015, Rodrigues et al 2016. Indeed simultaneous quantile fitting is a challenging problem that has gained a lot of attention in recent times.…”

“…This fact reduces the problem of infinite quantile monotonicity constraints to ensuring monotonicity only at the vertices of the convex set. Rodrigues et al (2016) used quantile pyramids at selected vertices of the convex hull of the predictor cloud, non-crossing conditions being checked at the remaining vertices. However, in practice, convex hulls of datapoints in dimensions higher than 9 or 10 can be difficult to obtain, and when an algorithm is successful at doing so it may return a large number of vertices.…”

“…Quantile pyramid is a method for constructing a random probability measure via the quantile function. We will use hereafter these priors for simultaneous inference on quantile spline curves, in the spirit of the linear quantile regression model recently proposed in Rodrigues et al (2016).…”

“…Pyramid quantile regression was first presented in Rodrigues et al (2016) as an alternative Bayesian procedure for simultaneous quantile fitting. This approach compared favourably with competing methods in empirical studies.…”

“…Section 2 presents the optimal convex set enclosing the data cloud with the given number of vertices. Section 3 recalls some basics on quantile pyramids introduced by Hjort and Walker (2009) and briefly describe their use in Rodrigues et al (2016) in a regression set-up. In Section 4 we introduce the penalised quantile splines model and describe the modelling set up.…”