Quantile regression for count data with optimized jittering

Quantile regression (QR) is a natural extension to the classic linear regression. It models the conditional quantiles of the continuous response variable instead of modeling the conditional mean. Often times we encounter discrete response variables, such as counts or other kind of categorical variables. Due to the discontinuity of counts, traditional QR creates systematic bias when applied to count data and hence is not directly applicable. Jittering with uniform random perturbations is one of the options to smooth the discrete response. In this thesis, we propose a new QR model for count data, which improves the existing uniform jittering method. The proposed approach involves artificially adding Tweedie or Beta random perturbations to original count response, generating pseudo-continuous QR response. Through proper selection of perturbation parameters, jittering can provide better parameter estimation of the regression parameters as compared with the existing methods. We employ the Asymmetric Laplace Distribution (ALD) to determine the optimal perturbation parameters through the Monte-Carlo Expectation Maximization (MCEM) and Metropolis-Hasting (MH) sampler. Our proposed method for QR model provides consistent estimators of the QR coefficients. The estimators follow asymptotic normal distribution as sample size goes to infinity. Simulation studies show much improved performance when sample sizes are small to moderate. As an illustration, the proposed method was applied to analyze a fishery data.