Understanding Regression Heterogeneity Through Joint Estimation of Conditional Quantiles

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In many data-driven scientific investigations, the primary goal is to understand the relationship between a response variable and a set of predictors. Standard statistical techniques attempt to detect and quantify the nature of such relationships through changes in the average response. However, in the real world, predictor-response relationships are often more complex and nuanced. In disciplines like climate science, ecology, economics, public health and sociology, investigators are often interested in understanding changes to the extreme response percentiles. Additional insights are gained by quantifying how the rate of change varies as one moves from the average to the extremes. This project aims to develop sophisticated and theoretically sound statistical tools that can answer these questions from large and complex data sets. Statistical tools are sought within the recently popularized modeling framework of linear quantile regression. The proposed framework expands the scope of linear quantile regression to scenarios where the response variables exhibit additional dependency. Such dependency manifests in many common situations, e.g., when one simultaneously measures multiple response variables per observation unit, when a response is measured repeatedly over time, or, when data is drawn from a network of individuals. Standing between the promise of quantile regression and its wider applicability is the lack of a proper model-based estimation framework. The PI has recently introduced a modeling framework that leads to Bayesian or penalized likelihood based joint estimation of linear quantile planes over arbitrary predictor spaces. Proposed model extensions augment this framework with autoregressive and copula formulations to address various kinds of structural dependency between the observation units. The project will develop efficient inference algorithms using advanced Bayesian techniques based on stochastic computation, and public, open source software in the form of R packages. Software development will incorporate possible leveraging of distributed computing architectures to render scalability to big data. For all model extensions, the PI will also carry out sharp analyses of theoretical guarantees on model performance by working out the large sample distribution theory of Bayesian parameter estimates.