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Modern applications require methods that are computationally feasible on large datasets but also preserve statistical efficiency. Frequently, these two concerns are seen as contradictory: approximation methods that enable computation are assumed to degrade statistical performance relative to exact methods. In applied mathematics, where much of the current theoretical work on approximation resides, the inputs are considered to be observed exactly. The prevailing philosophy is that while the exact problem is, regrettably, unsolvable, any approximation should be as small as possible. However, from a statistical perspective, an approximate or regularized solution may be preferable to the exact one. Regularization formalizes a trade-off between fidelity to the data and adherence to prior knowledge about the data-generating process such as smoothness or sparsity. The resulting estimator tends to be more useful, interpretable, and suitable as an input to other methods. In this paper, we propose new methodology for estimation and prediction under a linear model borrowing insights from the approximation literature. We explore these procedures from a statistical perspective and find that in many cases they improve both computational and statistical performance.
Model fitting often aims to fit a single model, assuming that the imposed form of the model is correct. However, there may be multiple possible underlying explanatory patterns in a set of predictors that could explain a response. Model selection with
Quadratic regression goes beyond the linear model by simultaneously including main effects and interactions between the covariates. The problem of interaction estimation in high dimensional quadratic regression has received extensive attention in the
We consider penalized regression models under a unified framework where the particular method is determined by the form of the penalty term. We propose a fully Bayesian approach that incorporates both sparse and dense settings and show how to use a t
We introduce a new approach to a linear-circular regression problem that relates multiple linear predictors to a circular response. We follow a modeling approach of a wrapped normal distribution that describes angular variables and angular distributi
Field observations form the basis of many scientific studies, especially in ecological and social sciences. Despite efforts to conduct such surveys in a standardized way, observations can be prone to systematic measurement errors. The removal of syst