Andreas Buja and Wolfgang Rolke (Work In Progress), Calibration for Simultaneity: (Re)sampling Methods for Simultaneous Inference with Applications to Function Estimation and Functional Data.
Abstract: We survey and illustrate a Monte Carlo technique for carrying out simple simultaneous inference with arbitrarily many statistics. Special cases of the technique have appeared in the literature, but there exists widespread unawareness of the simplicity and broad applicability of this solution to simultaneous inference.
The technique, here called “calibration for simultaneity" or CfS , consists of 1) limiting the search for coverage regions to a one-parameter family of nested regions, and 2) selecting from the family that region whose estimated coverage probability has the desired value. Natural one-parameter families are almost always available.
CfS applies whenever inference is based on a single distribution, for example: 1) fixed distributions such as Gaussians when diagnosing distributional assumptions, 2) conditional null distributions in exact tests with Neyman structure, in particular permutation tests, 3) bootstrap distributions for bootstrap standard error bands, 4) Bayesian posterior distributions for high-dimensional posterior probability regions, or 5) predictive distributions for multiple prediction intervals.
CfS is particularly useful for estimation of any type of function, such as empirical Q-Q curves, empirical CDFs, density estimates, smooths, generally any type of _t, and functions estimated from functional data.
A special case of CfS is equivalent to p-value adjustment (Westfall and Young, 1993). Conversely, the notion of a p-value can be extended to any simultaneous coverage problem that is solved with a one-parameter family of coverage regions.
Arun Kumar Kuchibhotla, Lawrence D. Brown, Andreas Buja, Edward I. George, Linda Zhao (2021), Uniform-in-Submodel Bounds for Linear Regression in a Model Free Framework, Econometric Theory, (in press) ().
Arun Kumar Kuchibhotla, Lawrence D. Brown, Andreas Buja, Edward I. George, Linda Zhao (2020), A Model Free Perspective for Linear Regression: Uniform-in-model Bounds for Post Selection Inference, Econometric Theory, (to appear) ().
Abstract: For the last two decades, high-dimensional data and methods have proliferated
throughout the literature. The classical technique of linear regression, however, has not lost
its touch in applications. Most high-dimensional estimation techniques can be seen as variable
selection tools which lead to a smaller set of variables where classical linear regression
technique applies. In this paper, we prove estimation error and linear representation bounds
for the linear regression estimator uniformly over (many) subsets of variables. Based on deterministic
inequalities, our results provide “good” rates when applied to both independent
and dependent data. These results are useful in correctly interpreting the linear regression
estimator obtained after exploring the data and also in post model-selection inference. All
the results are derived under no model assumptions and are non-asymptotic in nature.
Richard A. Berk, Andreas Buja, Lawrence D. Brown, Edward I. George, Arun Kumar Kuchibhotla, Weijie Su, Linda Zhao (2020), Assumption Lean Regression, American Statistician, (in press) ().
Richard A. Berk, Matthew Olson, Andreas Buja, Aurelie Ouss (2020), Using Recursive Partitioning to Find and Estimate Heterogenous Treatment Effects in Randomized Clinical Trials, Journal of Experimential Criminology, (to appear) ().
Arun Kumar Kuchibhotla, Lawrence D. Brown, Andreas Buja, Edward I. George, Linda Zhao (2020), Valid Post-selection Inference in Model-free Linear Regression, Annals of Statistics, 48 (5), pp. 2953-2981.
Abstract: This paper provides multiple approaches to perform valid post-selection inference in an assumption-lean regression analysis. To the best of our knowledge, this is the first work that provides valid post-selection inference for regression analysis in such a general settings that include independent, m-dependent random variables.
Andreas Buja, Arun Kumar Kuchibhotla, Richard A. Berk, Edward I. George, Eric Tchetgen Tchetgen, Linda Zhao (2020), Models as Approximations—Rejoinder, Statistical Science, 34 (4), pp. 606-620.
Andreas Buja, Lawrence D. Brown, Richard A. Berk, Edward I. George, Emil Pitkin, Mikhail Traskin, Kai Zhang, Linda Zhao (2019), Models as Approximations I: Consequences Illustrated with Linear Regression, Statistical Science, 34 (4), pp. 523-544.
Andreas Buja, Lawrence D. Brown, Arun Kumar Kuchibhotla, Richard A. Berk, Edward I. George, Linda Zhao (2019), Models as Approximations II: A Model-Free Theory of Parametric Regression, Statistical Science, 34 (4), pp. 345-365.
Arun Kumar Kuchibhotla, Lawrence D. Brown, Andreas Buja, Junhui Cai (Working), All of Linear Regression.
