Suppose data consist of a random sample from a distribution function $F_Y$, which is unknown, and that interest focuses on inferences on $\bft$, a vector of quantiles of $F_Y$. When the likelihood function is not fully specified, a posterior density cannot be calculated and Bayesian inference is difficult. This article considers an approach which relies on a substitution likelihood characterized by a vector of quantiles. Properties of the substitution likelihood are investigated, strategies for prior elicitation are presented, and a general framework is proposed for quantile regression modeling. Posterior computation proceeds via a Metropolis algorithm that utilizes a normal approximation to the posterior. Results from a simulation study are presented, and the methods are illustrated through application to data from a genotoxicity experiment.

Keywords: Comet assay; Nonparametric; Median regression; Order constraints; Prior elicitation; Quantile regression; Single cell electrophoresis; Substitution likelihood.

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