  Despite the increasing popularity of quantile regression models for
continuous responses, models for count data have so far received little
attention. The main quantile regression technique for count data involves
adding uniform random noise or "jittering", thus overcoming the problem that
the conditional quantile function is not a continuous function of the
parameters of interest. Although jittering allows estimating the conditional
quantiles, it has the drawback that, for small values of the response variable
$Y,$ the added noise can have a large influence on the estimated quantiles. In
addition, quantile regression can lead to "crossing" quantiles. We propose a
Bayesian Dirichlet process (DP)-based approach to quantile regression for count
data. The approach is based on an adaptive DP mixture (DPM) of COM-Poisson
regression models and determines the quantiles by estimating the density of the
data, thus eliminating all the aforementioned problems. Taking advantage of the
exchange algorithm, the proposed MCMC algorithm can be applied to distributions
on which the likelihood can only be computed up to a normalising constant.
