Mid-quantile regression for discrete responses
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We develop quantile regression methods for discrete responses by extending Parzen's definition of marginal mid-quantiles. As opposed to existing approaches, which are based on either jittering or latent constructs, we use interpolation and define the conditional mid-quantile function as the inverse of the conditional mid-distribution function. We propose a two-step estimator whereby, in the first step, conditional mid-probabilities are obtained nonparametrically and, in the second step, regression coefficients are estimated by solving an implicit equation. When constraining the quantile index to a data-driven admissible range, the second-step estimating equation has a least-squares type, closed-form solution. The proposed estimator is shown to be strongly consistent and asymptotically normal. A simulation study and real data applications are presented. Our methods can be applied to a large variety of discrete responses, including binary, ordinal, and count variables.
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