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1. Nonparametric Estimation of Uncertainty Sets for Robust Optimization

   https://doi.org/10.1109/CDC42340.2020.9303863  

   Alexeenko, Polina; Bitar, Eilyan (January 2021, 2020 59th IEEE Conference on Decision and Control (CDC))

   null (Ed.)

   We investigate a data-driven approach to constructing uncertainty sets for robust optimization problems, where the uncertain problem parameters are modeled as random variables whose joint probability distribution is not known. Relying only on independent samples drawn from this distribution, we provide a nonparametric method to estimate uncertainty sets whose probability mass is guaranteed to approximate a given target mass within a given tolerance with high confidence. The nonparametric estimators that we consider are also shown to obey distribution-free finite-sample performance bounds that imply their convergence in probability to the given target mass. In addition to being efficient to compute, the proposed estimators result in uncertainty sets that yield computationally tractable robust optimization problems for a large family of constraint functions. 

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2. Counterfactual Sensitivity and Robustness

   https://doi.org/10.3982/ECTA17232  

   Christensen, Timothy; Connault, Benjamin (January 2023, Econometrica)

   We propose a framework for analyzing the sensitivity of counterfactuals to parametric assumptions about the distribution of latent variables in structural models. In particular, we derive bounds on counterfactuals as the distribution of latent variables spans nonparametric neighborhoods of a given parametric specification while other “structural” features of the model are maintained. Our approach recasts the infinite‐dimensional problem of optimizing the counterfactual with respect to the distribution of latent variables (subject to model constraints) as a finite‐dimensional convex program. We also develop an MPEC version of our method to further simplify computation in models with endogenous parameters (e.g., value functions) defined by equilibrium constraints. We propose plug‐in estimators of the bounds and two methods for inference. We also show that our bounds converge to the sharp nonparametric bounds on counterfactuals as the neighborhood size becomes large. To illustrate the broad applicability of our procedure, we present empirical applications to matching models with transferable utility and dynamic discrete choice models. 

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3. Risk bounds for quantile trend filtering

   https://doi.org/10.1093/biomet/asab045  

   Madrid Padilla, Oscar Hernan; Chatterjee, Sabyasachi (September 2021, Biometrika)

   Summary We study quantile trend filtering, a recently proposed method for nonparametric quantile regression, with the goal of generalizing existing risk bounds for the usual trend-filtering estimators that perform mean regression. We study both the penalized and the constrained versions, of order $$r \geqslant 1$$, of univariate quantile trend filtering. Our results show that both the constrained and the penalized versions of order $$r \geqslant 1$$ attain the minimax rate up to logarithmic factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded-variation signals. Moreover, we show that if the true vector of quantiles is a discrete spline with a few polynomial pieces, then both versions attain a near-parametric rate of convergence. Corresponding results for the usual trend-filtering estimators are known to hold only when the errors are sub-Gaussian. In contrast, our risk bounds are shown to hold under minimal assumptions on the error variables. In particular, no moment assumptions are needed and our results hold under heavy-tailed errors. Our proof techniques are general, and thus can potentially be used to study other nonparametric quantile regression methods. To illustrate this generality, we employ our proof techniques to obtain new results for multivariate quantile total-variation denoising and high-dimensional quantile linear regression. 

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4. UNIFORM-IN-SUBMODEL BOUNDS FOR LINEAR REGRESSION IN A MODEL-FREE FRAMEWORK

   https://doi.org/10.1017/S0266466621000219  

   Kuchibhotla, Arun K.; Brown, Lawrence D.; Buja, Andreas; George, Edward I.; Zhao, Linda (June 2021, Econometric Theory)

   For the last two decades, high-dimensional data and methods have proliferated throughout the literature. Yet, the classical technique of linear regression has not lost its usefulness in applications. In fact, many high-dimensional estimation techniques can be seen as variable selection that leads to a smaller set of variables (a “submodel”) where classical linear regression applies. We analyze linear regression estimators resulting from model selection by proving estimation error and linear representation bounds uniformly over sets of submodels. Based on deterministic inequalities, our results provide “good” rates when applied to both independent and dependent data. These results are useful in meaningfully interpreting the linear regression estimator obtained after exploring and reducing the variables and also in justifying post-model-selection inference. All results are derived under no model assumptions and are nonasymptotic in nature. 

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5. On the maximal deviation of kernel regression estimators with NMAR response variables

   https://doi.org/10.1007/s00362-022-01293-0  

   Mojirsheibani, Majid (February 2022, Statistical Papers)

   This article focuses on the problem of kernel regression estimation in the presence of nonignorable incomplete data with particular focus on the limiting distribution of the maximal deviation of the proposed estimators. From an applied point of view, such a limiting distribution enables one to construct asymptotically correct uniform bands, or perform tests of hypotheses, for a regression curve when the available data set suffers from missing (not necessarily at random) response values. Furthermore, such asymptotic results have always been of theoretical interest in mathematical statistics. We also present some numerical results that further confirm and complement the theoretical developments of this paper. 

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