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A Bayesian nonparametric approach to inference for quantile regression
 Journal of Business and Economic Statistics
, 2009
"... In several regression applications, a dierent structural relationship might be anticipated for the higher or lower responses than the average responses. In such cases, quantile regression analysis can uncover important features that would likely be overlooked by traditional mean regression. We devel ..."
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Cited by 14 (8 self)
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In several regression applications, a dierent structural relationship might be anticipated for the higher or lower responses than the average responses. In such cases, quantile regression analysis can uncover important features that would likely be overlooked by traditional mean regression. We develop a Bayesian method for fully nonparametric modelbased quantile regression. The approach involves exible Dirichlet process mixture models for the joint distribution of the response and the covariates, with posterior inference for dierent quantile curves emerging from the conditional distribution of the response given the covariates. Inference is implemented using a combination of posterior simulation methods for Dirichlet process mixtures. Partially observed responses can also be handled within the proposed modeling framework leading to a novel nonparametric method for Tobit quantile regression. We use simulated data sets as well as two data examples from the literature to illustrate the utility of the model, in particular, its capacity to uncover nonlinearities in quantile regression curves as well as nonstandard features in the response distribution.
Bayesian quantile regression methods
 Journal of Applied Econometrics
"... This paper is a study of the application of Bayesian Exponentially Tilted Empirical Likelihood to inference about quantile regressions. In the case of simple quantiles we show the exact form for the likelihood implied by this method and compare it with the Bayesian bootstrap and with Jeffreys ’ meth ..."
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Cited by 10 (0 self)
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This paper is a study of the application of Bayesian Exponentially Tilted Empirical Likelihood to inference about quantile regressions. In the case of simple quantiles we show the exact form for the likelihood implied by this method and compare it with the Bayesian bootstrap and with Jeffreys ’ method. For regression quantiles we derive the asymptotic form of the posterior density. We also examine MCMC simulations with a proposal density formed from an overdispersed version of the limiting normal density. We show that the algorithm works well even in models with an endogenous regressor when the instruments are not too weak.
Simultaneous linear quantile regression: A semiparametric bayesian approach
 In press
, 2010
"... We introduce a semiparametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computationally challenging due to the associated monotonicity constraint ..."
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Cited by 8 (0 self)
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We introduce a semiparametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computationally challenging due to the associated monotonicity constraint on the quantile curves. For a univariate covariate, we present a simpler equivalent characterization of the monotonicity constraint through an interpolation of two monotone curves. The resulting formulation leads to a tractable likelihood function and is embedded within a Bayesian framework where the two monotone curves are modeled via logistic transformations of a smooth Gaussian process. A multivariate extension is proposed by combining the full support univariate model with a linear projection of the predictors. The resulting singleindex model remains easy to fit and provides substantial and measurable improvement over the first order linear heteroscedastic model. Two illustrative applications of the proposed method are provided.
Spatial quantile multiple regression using the asymmetric laplace process
 Bayesian Analysis
, 2012
"... Abstract. We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process ( ..."
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Cited by 7 (2 self)
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Abstract. We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process (ALP) for quantile regression with spatially dependent errors. By taking advantage of a convenient conditionally Gaussian representation of the asymmetric Laplace distribution, we are able to straightforwardly incorporate spatial dependence in this process. We develop the properties of this process under several specifications, each of which induces different smoothness and covariance behavior at the extreme quantiles. We demonstrate the advantages that may be gained by incorporating spatial dependence into this conditional quantile model by applying it to a data set of log selling prices of homes in Baton Rouge, LA, given characteristics of each house. We also introduce the asymmetric Laplace predictive process (ALPP) which accommodates large data sets, and apply it to a data set of birth weights given maternal covariates for several thousand births in North Carolina in 2000. By modeling the spatial structure in the data, we are able to show, using a check loss function, improved performance on each of the data sets for each of the quantiles at which the model was fit.
A Partially collapsed Gibbs sampler for Bayesian quantile regression
, 2009
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Bayesian quantile regression for censored data. Biometrics
, 2013
"... Summary: In this paper we propose a semiparametric quantile regression model for censored survival data. Quantile regression permits covariates to affect survival differently at different stages in the followup period, thus providing a comprehensive study of the survival distribution. We take a sem ..."
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Summary: In this paper we propose a semiparametric quantile regression model for censored survival data. Quantile regression permits covariates to affect survival differently at different stages in the followup period, thus providing a comprehensive study of the survival distribution. We take a semiparametric approach, representing the quantile process as a linear combination of basis functions. The basis functions are chosen so that the prior for the quantile process is centered on a simple locationscale model, but flexible enough to accommodate a wide range of quantile processes. We show in a simulation study that this approach is competitive with existing methods. The method is illustrated using data from a drug treatment study, where we find that the Bayesian model often gives smaller measures of uncertainty than its competitors, and thus identifies more significant effects. Key words: Accelerated failure time model; Markov chain Monte Carlo; Quantile Regression; Survival data. This paper has been submitted for consideration for publication in Biometrics Bayesian quantile regression for censored data 1 1.
Estimating the Health Impact of Climate Change with Calibrated Climate Model Output
, 2012
"... Studies on the health impacts of climate change routinely use climate model output as future exposure projection. Uncertainty quantification, usually in the form of sensitivity analysis, has focused predominantly on the variability arise from different emission scenarios or multimodel ensembles. Th ..."
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Cited by 2 (2 self)
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Studies on the health impacts of climate change routinely use climate model output as future exposure projection. Uncertainty quantification, usually in the form of sensitivity analysis, has focused predominantly on the variability arise from different emission scenarios or multimodel ensembles. This paper describes a Bayesian spatial quantile regression approach to calibrate climate model output for examining to the risks of future temperature on adverse health outcomes. Specifically, we first estimate the spatial quantile process for climate model output using nonlinear monotonic regression during a historical period. The quantile process is then calibrated using the quantile functions estimated from the observed monitoring data. Our model also downscales the gridded climate model output to the pointlevel for projecting future exposure over a specific geographical region. The quantile regression approach is motivated by the need to better characterize the tails of future temperature distribution where the greatest health impacts are likely to occur. We applied the methodology to calibrate temperature projections from a regional climate model for the period 2041 to 2050. Accounting for calibration uncertainty, we calculated the number of of excess deaths attributed to future temperature for three cities in the US state of Alabama.
Modelbased approaches to nonparametric Bayesian quantile regression
"... In several regression applications, a different structural relationship might be anticipated for the higher or lower responses than the average responses. In such cases, quantile regression analysis can uncover important features that would likely be overlooked by mean regression. We develop two dis ..."
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Cited by 1 (1 self)
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In several regression applications, a different structural relationship might be anticipated for the higher or lower responses than the average responses. In such cases, quantile regression analysis can uncover important features that would likely be overlooked by mean regression. We develop two distinct Bayesian approaches to fully nonparametric modelbased quantile regression. The first approach utilizes an additive regression framework with Gaussian process priors for the quantile regression functions and a scale uniform Dirichlet process mixture prior for the error distribution, which yields flexible unimodal error density shapes. Under the second approach, the joint distribution of the response and the covariates is modeled with a Dirichlet process mixture of multivariate normals, with posterior inference for different quantile curves emerging through the conditional distribution of the response given the covariates. The proposed nonparametric prior probability models allow the data to uncover nonlinearities in the quantile regression function and nonstandard distributional features in the response distribution. Inference is implemented using a combination of posterior simulation methods for Dirichlet process mixtures. We illustrate the performance of the proposed models using simulated and real data sets.
Calibration of numerical model output using nonparametric
, 2011
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Statistical Modelling 2013; 13(3): 223–252 Bayesian semiparametric additive quantile regression
, 2013
"... Abstract: Quantile regression provides a convenient framework for analyzing the impact of covariates on the complete conditional distribution of a response variable instead of only the mean. While frequentist treatments of quantile regression are typically completely nonparametric, a Bayesian formu ..."
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Abstract: Quantile regression provides a convenient framework for analyzing the impact of covariates on the complete conditional distribution of a response variable instead of only the mean. While frequentist treatments of quantile regression are typically completely nonparametric, a Bayesian formulation relies on assuming the asymmetric Laplace distribution as auxiliary error distribution that yields posterior modes equivalent to frequentist estimates. In this paper, we utilize a locationscale mixture of normals representation of the asymmetric Laplace distribution to transfer different flexible modelling concepts from Gaussian mean regression to Bayesian semiparametric quantile regression. In particular, we will consider highdimensional geoadditive models comprising LASSO regularization priors and mixed models with potentially nonnormal random effects distribution modeled via a Dirichlet process mixture. These extensions are illustrated using two largescale applications on net rents in Munich and longitudinal measurements on obesity among children. The impact of the likelihood misspecification that underlies the Bayesian formulation of quantile regression is studied in terms of simulations.