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Abstract:
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In spite of its great promise, linear quantile regression is difficult to apply beyond the IID setting, due largely to the limitations originating from its model-free nature. It is time to upgrade traditional linear quantile regression to overarching linear quantile models of heterogeneous predictor effects that offer fully generative probabilistic models for the data. We discuss such a linear quantile model to adjust for noise correlation in point-referenced spatial data without replications. We provide ample evidence that such adjustments improve parameter estimation, uncertainty quantification and prediction. As an extension of the Bayesian spatial random effects model, the proposed joint spatial quantile regression (JSQR) model offers a comprehensive and practicable solution to the problem of spatial regression beyond the mean. Through two case studies we show that JSQR produces interpretable estimates of potentially heterogeneous predictor effects, offers excellent fit to hold-out data and can successfully adapt to heavy tailed response and tail dependence.
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