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Full text release has been delayed at the author's request until August 04, 2026

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Bayesian Quantile Regression via Dependent Quantile Pyramids

An, Hyoin
http://rave.ohiolink.edu/etdc/view?acc_num=osu1717623636412423

Abstract Details

2024, Doctor of Philosophy, Ohio State University, Statistics.
Quantile regression (QR) has drawn increased attention as an attractive alternative to mean regression. QR was motivated by the realization that extreme quantiles often have a different relationship with covariates than do the centers of the response distributions. QR can target quantiles in the tail of the response distribution and provide additional insights into the response distribution. QR also tends to be more robust to outliers than is mean regression. A fundamental property of quantiles is their monotonicity, ensuring that they increase with higher quantile levels. When analyzing multiple quantiles of the response distribution, fitting individual quantile regressions may not guarantee the correct ordering of the conditional quantiles. If the ordering of the quantiles is not maintained, the quantile regression curves cross one another. To mitigate the crossing quantile issue, the idea of simultaneous quantile regression (SQR) has been introduced. SQR aims to fit multiple quantile regression curves under the monotonocity constraint. With SQR, we can understand the relationship between covariates and multiple response quantiles in a relatively comprehensive way. However, despite the practicality of SQR, obtaining multiple QR curves, especially in a flexible, nonlinear form, continues to be challenging. We propose a new class of stochastic processes, a process of dependent quantile pyramids (DQP). This class is applied to build a flexible SQR model that falls within the Bayesian nonparametric framework. The DQP generalizes the quantile pyramid, a model for a single set of quantiles without a covariate. The generalization replaces each scalar variate in the quantile pyramid with a stochastic process whose index set is a covariate space. The resulting model is a distribution-valued stochastic process which provides a nonparametric distribution at each value of the covariate. We rigorously establish the existence of the model and describe several of its properties. The DQP allows for nonlinear QR and automatically ensures noncrossing of QR curves on the covariate space. To our knowledge, there is no other Bayesian approach that can directly model the conditional quantiles of interest in a flexible fashion. Next, we develop a canonical construction of the proposed method for practical implementation. A detailed guide is provided about how to build and control each component of the model. We also note that a posterior inference can be made separately from modeling. We first fit our highly flexible model to extract the information from the data, and then make simplified inference to facilitate straightforward understanding. Simulation studies evaluate the performance of our model and show that it is robust and competitive. The model is applied to cyclone intensity analysis, providing results that are consistent with those in the literature and sharpening insights from the data. We also conduct a sensitivity analysis for the model to see the impact of the number of pre-specified quantiles. Finally, we introduce variations of our model, tailored to various needs. One of the variations involves limiting the number of pyramids on a covariate space and interpolating conditional quantiles between these pyramids. This approach can be helpful for updating pyramids in Markov Chain Monte Carlo procedures, particularly in cases involving continuous covariates, while also reducing the computational overhead associated with pyramid construction. On the other hand, we might suspect that some covariates impact the center and spread of the response distribution but do not affect its shape. For such cases, we present a location-scale family of DQP. This method lets us build dependent quantile pyramids in a lower dimensional space, where the shape of the distribution is believed to change, while allowing higher dimensional covariates to affect the location and scale of the conditional quantiles. This approach can not only incorporate knowledge regarding the impact of covariates on the response distribution, but also improve computational efficiency. For smoother estimation of the response density, we suggest a smooth interpolation of conditional quantiles at a given covariate value.
Steven MacEachern (Advisor)
Mario Peruggia (Committee Member)
Oksana Chkrebtii (Committee Member)
Statistics
Quantile regression; Noncrossing quantiles; Bayesian nonparametrics; Bayesian inference; Quantile pyramids; Gaussian processes; Prokhorov metric

Recommended Citations

Citations

  • An, H. (2024). Bayesian Quantile Regression via Dependent Quantile Pyramids [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1717623636412423

    APA Style (7th edition)

  • An, Hyoin. Bayesian Quantile Regression via Dependent Quantile Pyramids. 2024. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1717623636412423.

    MLA Style (8th edition)

  • An, Hyoin. "Bayesian Quantile Regression via Dependent Quantile Pyramids." Doctoral dissertation, Ohio State University, 2024. http://rave.ohiolink.edu/etdc/view?acc_num=osu1717623636412423

    Chicago Manual of Style (17th edition)

osu1717623636412423
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