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Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. For analysing spatially indexed data, we address this difficulty by generalizing the joint quantile regression model of Yang and Tokdar (Journal of the American Statistical Association, 2017, 112(519), 1107–1120) and characterizing spatial dependence via a Gaussian or t-copula process on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out spatial quantile smoothing. An effective model comparison criteria is provided, particularly for selecting between different model specifications of tail heaviness and tail dependence. Extensive simulation studies and two real applications to particulate matter concentration and wildfire risk are presented to illustrate substantial gains in inference quality, prediction accuracy and uncertainty quantification over existing alternatives.

 

A novel statistical method is proposed and investigated for estimating a heavy tailed density under mildsmoothness assumptions. Statistical analyses of heavy-tailed distributions are susceptible to the problem ofsparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk.The proposed Bayesian method avoids this problem by incorporating smoothness and tail regularizationthrough a carefully specified semiparametric prior distribution, and is able to consistently estimate boththe density function and its tail index at near minimax optimal rates of contraction. A joint, likelihood drivenestimation of the bulk and the tail is shown to help improve uncertainty assessment in estimating the tailindex parameter and offer more accurate and reliable estimates of the high tail quantiles compared tothresholding methods. Supplementary materials for this article are available online.