  We propose a new general approach for estimating the effect of a binary
treatment on a continuous and potentially highly skewed response variable, the
generalized quantile treatment effect (GQTE). The GQTE is defined as the
difference between a function of the quantiles under the two treatment
conditions. As such, it represents a generalization over the standard
approaches typically used for estimating a treatment effect (i.e., the average
treatment effect and the quantile treatment effect) because it allows the
comparison of any arbitrary characteristic of the outcome's distribution under
the two treatments. Following Dominici et al. (2005), we assume that a
pre-specified transformation of the two quantiles is modeled as a smooth
function of the percentiles. This assumption allows us to link the two quantile
functions and thus to borrow information from one distribution to the other.
The main theoretical contribution we provide is the analytical derivation of a
closed form expression for the likelihood of the model. Exploiting this result
we propose a novel Bayesian inferential methodology for the GQTE. We show some
finite sample properties of our approach through a simulation study which
confirms that in some cases it performs better than other nonparametric
methods. As an illustration we finally apply our methodology to the 1987
National Medicare Expenditure Survey data to estimate the difference in the
single hospitalization medical cost distributions between cases (i.e., subjects
affected by smoking attributable diseases) and controls.
