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1. A Bias Bound Approach to Non-parametric Inference

   https://doi.org/10.1093/restud/rdz065  

   Schennach, Susanne M (January 2020, The Review of Economic Studies)

   Abstract The traditional approach to obtain valid confidence intervals for non-parametric quantities is to select a smoothing parameter such that the bias of the estimator is negligible relative to its standard deviation. While this approach is apparently simple, it has two drawbacks: first, the question of optimal bandwidth selection is no longer well-defined, as it is not clear what ratio of bias to standard deviation should be considered negligible. Second, since the bandwidth choice necessarily deviates from the optimal (mean squares-minimizing) bandwidth, such a confidence interval is very inefficient. To address these issues, we construct valid confidence intervals that account for the presence of a non-negligible bias and thus make it possible to perform inference with optimal mean squared error minimizing bandwidths. The key difficulty in achieving this involves finding a strict, yet feasible, bound on the bias of a non-parametric estimator. It is well-known that it is not possible to consistently estimate the pointwise bias of an optimal non-parametric estimator (for otherwise, one could subtract it and obtain a faster convergence rate violating Stone’s bounds on the optimal convergence rates). Nevertheless, we find that, under minimal primitive assumptions, it is possible to consistently estimate an upper bound on the magnitude of the bias, which is sufficient to deliver a valid confidence interval whose length decreases at the optimal rate and which does not contradict Stone’s results. 

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2. Optimal bandwidth choice for robust bias-corrected inference in regression discontinuity designs

   https://doi.org/10.1093/ectj/utz022  

   Calonico, Sebastian; Cattaneo, Matias D; Farrell, Max H (November 2019, The Econometrics Journal)

   Summary Modern empirical work in regression discontinuity (RD) designs often employs local polynomial estimation and inference with a mean square error (MSE) optimal bandwidth choice. This bandwidth yields an MSE-optimal RD treatment effect estimator, but is by construction invalid for inference. Robust bias-corrected (RBC) inference methods are valid when using the MSE-optimal bandwidth, but we show that they yield suboptimal confidence intervals in terms of coverage error. We establish valid coverage error expansions for RBC confidence interval estimators and use these results to propose new inference-optimal bandwidth choices for forming these intervals. We find that the standard MSE-optimal bandwidth for the RD point estimator is too large when the goal is to construct RBC confidence intervals with the smaller coverage error rate. We further optimize the constant terms behind the coverage error to derive new optimal choices for the auxiliary bandwidth required for RBC inference. Our expansions also establish that RBC inference yields higher-order refinements (relative to traditional undersmoothing) in the context of RD designs. Our main results cover sharp and sharp kink RD designs under conditional heteroskedasticity, and we discuss extensions to fuzzy and other RD designs, clustered sampling, and pre-intervention covariates adjustments. The theoretical findings are illustrated with a Monte Carlo experiment and an empirical application, and the main methodological results are available in R and Stata packages. 

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3. Debiased inference for a covariate-adjusted regression function

   https://doi.org/10.1093/jrsssb/qkae041  

   Takatsu, Kenta; Westling, Ted (May 2024, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   In this article, we study nonparametric inference for a covariate-adjusted regression function. This parameter captures the average association between a continuous exposure and an outcome after adjusting for other covariates. Under certain causal conditions, it also corresponds to the average outcome had all units been assigned to a specific exposure level, known as the causal dose–response curve. We propose a debiased local linear estimator of the covariate-adjusted regression function and demonstrate that our estimator converges pointwise to a mean-zero normal limit distribution. We use this result to construct asymptotically valid confidence intervals for function values and differences thereof. In addition, we use approximation results for the distribution of the supremum of an empirical process to construct asymptotically valid uniform confidence bands. Our methods do not require undersmoothing, permit the use of data-adaptive estimators of nuisance functions, and our estimator attains the optimal rate of convergence for a twice differentiable regression function. We illustrate the practical performance of our estimator using numerical studies and an analysis of the effect of air pollution exposure on cardiovascular mortality. 

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4. Optimal Nonparametric Inference with Two-Scale Distributional Nearest Neighbors

   https://doi.org/10.1080/01621459.2022.2115375  

   Demirkaya, Emre; Fan, Yingying; Gao, Lan; Lv, Jinchi; Vossler, Patrick; Wang, Jingbo (October 2022, Journal of the American Statistical Association)

   The weighted nearest neighbors (WNN) estimator has been popularly used as a flexible and easy-to-implement nonparametric tool for mean regression estimation. The bagging technique is an elegant way to form WNN estimators with weights automatically generated to the nearest neighbors (Steele, 2009; Biau et al., 2010); we name the resulting estimator as the distributional nearest neighbors (DNN) for easy reference. Yet, there is a lack of distributional results for such estimator, limiting its application to statistical inference. Moreover, when the mean regression function has higher-order smoothness, DNN does not achieve the optimal nonparametric convergence rate, mainly because of the bias issue. In this work, we provide an in-depth technical analysis of the DNN, based on which we suggest a bias reduction approach for the DNN estimator by linearly combining two DNN estimators with different subsampling scales, resulting in the novel two-scale DNN (TDNN) estimator. The two-scale DNN estimator has an equivalent representation of WNN with weights admitting explicit forms and some being negative. We prove that, thanks to the use of negative weights, the two-scale DNN estimator enjoys the optimal nonparametric rate of convergence in estimating the regression function under the fourth order smoothness condition. We further go beyond estimation and establish that the DNN and two-scale DNN are both asymptotically normal as the subsampling scales and sample size diverge to infinity. For the practical implementation, we also provide variance estimators and a distribution estimator using the jackknife and bootstrap techniques for the two-scale DNN. These estimators can be exploited for constructing valid confidence intervals for nonparametric inference of the regression function. The theoretical results and appealing nite-sample performance of the suggested two-scale DNN method are illustrated with several simulation examples and a real data application. 

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5. Portfolio Selection Problem

   Begashaw, Negash; Comert, Gurcan; Medhin, Negash G (June 2025, Neural Parallel and Scientific Computations)

   Given a set of securities or assets it is of interest to find an optimal way of investing in these assets. What is optimal has to specified. The objective is to optimize the return consistent with the specified objective. When there are several assets it is unlikely all the assets will increase if they are correlated. It is necessary to diversify one’s assets for a secure return. To deal with the different assets a combination of the assets should be considered with constraints as needed. One approach is the Markowitz mean-variance model where the mean variance is minimized including constraints. In this paper neural networks and machine learning are used to extend the ways of dealing with portfolio asset allocation. Portfolio selection problem in an efficient way. The use of heuristic algorithms in this case is imperative. In the past some heuristic methods based mainly on evolutionary algorithms, tabu search and simulated annealing have been developed. The purpose of this paper is to consider a particular neural network model, the Hopfield network, which has been used to solve some other optimisation problems and apply it here to the portfolio selection problem, comparing the new results to those obtained with previous heuristic algorithms. Although great success has been achieved for portfolio analysis with the birth of Markowitz model, the demand for timely decision making has significantly increased especially in recent years with the advancement of high frequency trading (HFT), which combines powerful computing servers and the fastest Internet connection to trade at extremely high speeds. This demand poses new challenges to portfolio solvers for real-time processing in the face of time-varying parameters. Neural networks, as one of the most powerful machine learning tools has seen great progress in recent years for financial data analysis and signal processing ([1], [14]). Using computational methods, e.g., machine learning and data analytics, to empower conventional finance is becoming a trend widely adopted in leading investment companies ([3]). 

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