Arun Kumar Kuchibhotla

Bio: Arun Kumar Kuchibhotla is an academic researcher from Carnegie Mellon University. The author has contributed to research in topics: Estimator & Mathematics. The author has an hindex of 10, co-authored 45 publications receiving 327 citations. Previous affiliations of Arun Kumar Kuchibhotla include Indian Statistical Institute & University of Pennsylvania.

Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression

Abstract: Concentration inequalities form an essential toolkit in the study of high dimensional (HD) statistical methods. Most of the relevant statistics literature in this regard is based on sub-Gaussian or sub-exponential tail assumptions. In this paper, we first bring together various probabilistic inequalities for sums of independent random variables under much weaker exponential type (namely sub-Weibull) tail assumptions. These results extract a part sub-Gaussian tail behavior in finite samples, matching the asymptotics governed by the central limit theorem, and are compactly represented in terms of a new Orlicz quasi-norm - the Generalized Bernstein-Orlicz norm - that typifies such tail behaviors. We illustrate the usefulness of these inequalities through the analysis of four fundamental problems in HD statistics. In the first two problems, we study the rate of convergence of the sample covariance matrix in terms of the maximum elementwise norm and the maximum k-sub-matrix operator norm which are key quantities of interest in bootstrap, HD covariance matrix estimation and HD inference. The third example concerns the restricted eigenvalue condition, required in HD linear regression, which we verify for all sub-Weibull random vectors through a unified analysis, and also prove a more general result related to restricted strong convexity in the process. In the final example, we consider the Lasso estimator for linear regression and establish its rate of convergence under much weaker than usual tail assumptions (on the errors as well as the covariates), while also allowing for misspecified models and both fixed and random design. To our knowledge, these are the first such results for Lasso obtained in this generality. The common feature in all our results over all the examples is that the convergence rates under most exponential tails match the usual ones under sub-Gaussian assumptions.

Nested conformal prediction and quantile out-of-bag ensemble methods

Abstract: Conformal prediction is a popular tool for providing valid prediction sets for classification and regression problems, without relying on any distributional assumptions on the data. While the traditional description of conformal prediction starts with a nonconformity score, we provide an alternate (but equivalent) view that starts with a sequence of nested sets and calibrates them to find a valid prediction set. The nested framework subsumes all nonconformity scores, including recent proposals based on quantile regression and density estimation. While these ideas were originally derived based on sample splitting, our framework seamlessly extends them to other aggregation schemes like cross-conformal, jackknife+ and out-of-bag methods. We use the framework to derive a new algorithm (QOOB, pronounced cube) that combines four ideas: quantile regression, cross-conformalization, ensemble methods and out-of-bag predictions. We develop a computationally efficient implementation of cross-conformal, that is also used by QOOB. In a detailed numerical investigation, QOOB performs either the best or close to the best on all simulated and real datasets.

Models as Approximations II: A Model-Free Theory of Parametric Regression

Abstract: We develop a model-free theory of general types of parametric regression for i.i.d. observations. The theory replaces the parameters of parametric models with statistical functionals, to be called “regression functionals,” defined on large nonparametric classes of joint ${x\textrm{-}y}$ distributions, without assuming a correct model. Parametric models are reduced to heuristics to suggest plausible objective functions. An example of a regression functional is the vector of slopes of linear equations fitted by OLS to largely arbitrary ${x\textrm{-}y}$ distributions, without assuming a linear model (see Part I). More generally, regression functionals can be defined by minimizing objective functions, solving estimating equations, or with ad hoc constructions. In this framework, it is possible to achieve the following: (1) define a notion of “well-specification” for regression functionals that replaces the notion of correct specification of models, (2) propose a well-specification diagnostic for regression functionals based on reweighting distributions and data, (3) decompose sampling variability of regression functionals into two sources, one due to the conditional response distribution and another due to the regressor distribution interacting with misspecification, both of order $N^{-1/2}$, (4) exhibit plug-in/sandwich estimators of standard error as limit cases of ${x\textrm{-}y}$ bootstrap estimators, and (5) provide theoretical heuristics to indicate that ${x\textrm{-}y}$ bootstrap standard errors may generally be preferred over sandwich estimators.

A Model Free Perspective for Linear Regression: Uniform-in-model Bounds for Post Selection Inference

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

Valid post-selection inference in model-free linear regression

Abstract: Modern data-driven approaches to modeling make extensive use of covariate/model selection. Such selection incurs a cost: it invalidates classical statistical inference. A conservative remedy to the problem was proposed by Berk et al. (Ann. Statist. 41 (2013) 802–837) and further extended by Bachoc, Preinerstorfer and Steinberger (2016). These proposals, labeled “PoSI methods,” provide valid inference after arbitrary model selection. They are computationally NP-hard and have limitations in their theoretical justifications. We therefore propose computationally efficient confidence regions, named “UPoSI’ (“U” is for “uniform” or “universal.”) and prove large-$p$ asymptotics for them. We do this for linear OLS regression allowing misspecification of the normal linear model, for both fixed and random covariates, and for independent as well as some types of dependent data. We start by proving a general equivalence result for the post-selection inference problem and a simultaneous inference problem in a setting that strips inessential features still present in a related result of Berk et al. (Ann. Statist. 41 (2013) 802–837). We then construct valid PoSI confidence regions that are the first to have vastly improved computational efficiency in that the required computation times grow only quadratically rather than exponentially with the total number $p$ of covariates. These are also the first PoSI confidence regions with guaranteed asymptotic validity when the total number of covariates $p$ diverges (almost exponentially) with the sample size $n$. Under standard tail assumptions, we only require $(\log p)^{7}=o(n)$ and $k=o(\sqrt{n/\log p})$ where $k$ ($\le p$) is the largest number of covariates (model size) considered for selection. We study various properties of these confidence regions, including their Lebesgue measures, and compare them theoretically with those proposed previously.

Advanced Calculus I

Abstract: WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Fairness in Criminal Justice Risk Assessments: The State of the Art

Abstract: Objectives:Discussions of fairness in criminal justice risk assessments typically lack conceptual precision. Rhetoric too often substitutes for careful analysis. In this article, we seek to clarify...