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We introduce a new method of estimation of parameters in semiparametric and nonparametric models. The method employs U-statistics that are based on higher-order influence functions of the parameter of interest, which extend ordinary linear influence functions, and represent higher derivatives of this parameter. For parameters for which the representation cannot be perfect the method often leads to a bias-variance trade-off, and results in estimators that converge at a slower than √n-rate. In a number of examples, the resulting rate can be shown to be optimal. We are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity, where the parameter of interest cannot be estimated at √n-rate, but we also consider efficient √n-estimation using novel nonlinear estimators. The general approach is applied in detail to the example of estimating a mean response when the response is not always observed.

In some high-dimensional and semiparametric inference problems involving two populations, the parameter of interest can be characterized by two-sample U-statistics involving some nuisance parameters. In this work we first extend the framework of one-step estimation with cross-fitting to two-sample U-statistics, showing that using an orthogonalized influence function can effectively remove the first order bias, resulting in asymptotically normal estimates of the parameter of interest. As an example, we apply this method and theory to the problem of testing two-sample conditional distributions, also known as strong ignorability. When combined with a conformal-based rank-sum test, we discover that the nuisance parameters can be divided into two categories, where in one category the nuisance estimation accuracy does not affect the testing validity, whereas in the other the nuisance estimation accuracy must satisfy the usual requirement for the test to be valid. We believe these findings provide further insights into and enhance the conformal inference toolbox.

This article examines the role and the efficiency of nonconvex loss functions for binary classification problems. In particular, we investigate how to design adaptive and effective boosting algorithms that are robust to the presence of outliers in the data or to the presence of errors in the observed data labels. We demonstrate that nonconvex losses play an important role for prediction accuracy because of the diminishing gradient properties-the ability of the losses to efficiently adapt to the outlying data. We propose a new boosting framework called ArchBoost that uses diminishing gradient property directly and leads to boosting algorithms that are provably robust. Along with the ArchBoost framework, a family of nonconvex losses is proposed, which leads to the new robust boosting algorithms, named adaptive robust boosting (ARB). Furthermore, we develop a new breakdown point analysis and a new influence function analysis that demonstrate gains in robustness. Moreover, based only on local curvatures, we establish statistical and optimization properties of the proposed ArchBoost algorithms with highly nonconvex losses. Extensive numerical and real data examples illustrate theoretical properties and reveal advantages over the existing boosting methods when data are perturbed by an adversary or otherwise. Supplementary materials for this article are available online.

The efficient global optimization (EGO) algorithm is famous for its high efficiency in solving computationally expensive optimization problems. However, the expected improvement (EI) criterion used for picking up candidate points in the EGO process produces only one design point per optimization cycle, which is time-wasting when parallel computing can be used. In this work, a new criterion called pseudo expected improvement (PEI) is proposed for developing parallel EGO algorithms. In each cycle, the first updating point is selected by the initial EI function. After that, the PEI function is built to approximate the real updated EI function by multiplying the initial EI function by an influence function of the updating point. The influence function is designed to simulate the impact that the updating point will have on the EI function, and is only corresponding to the position of the updating point (not the function value of the updating point). Therefore, the next updating point can be identified by maximizing the PEI function without evaluating the first updating point. As the sequential process goes on, a desired number of updating points can be selected by the PEI criterion within one optimization cycle. The efficiency of the proposed PEI criterion is validated by six benchmarks with dimension from 2 to 6. The results show that the proposed PEI algorithm performs significantly better than the standard EGO algorithm, and gains significant improvements over five of the six test problems compared against a state-of-the-art parallel EGO algorithm. Furthermore, additional experiments show that it affects the convergence of the proposed algorithm significantly when the global maximum of the PEI function is not found. It is recommended to use as much evaluations as one can afford to find the global maximum of the PEI function.

Accurate prediction of dynamic mining subsidence is a critical factor in mitigating the disturbance of surface structures due to underground mining and therefore paramount for sustainable longwall mining practices. This paper presents an application of the Knothe influence function for predicting final and dynamic movements due to underground coal mining operations. Measured surface movements related to a single longwall panel located in the eastern U.S. are used to develop site-specific empirical parameters for subsidence prediction models with respect to final and dynamic movements in both the vertical and horizontal planes. Model calibration results for final deformations show that some of the empirical parameters derived for the specific site deviate from regional default values. Calibration tests for dynamic movements were conducted for points close to the panel edge, which are affected by the edge effect, and points in the middle of the panel, which are not. These yielded different time factor constants. Calculated deformations match measured data very well, which indicates that the Knothe formulation is applicable to modeling both final and dynamic deformations. The subsidence models developed for final and dynamic deformations were used to predict final and dynamic horizontal displacements at the site. The relative root mean square error of these predictions is rather high, which suggests that such predictions may be affected by other factors. However, specific portions of the prediction profiles display very low error. In addition, the displacement magnitudes are low, and the prediction model identifies the transitions in the displacement curves. Highlights The influence function method can be used to model final and dynamic subsidence and horizontal displacements above longwall operations in the eastern US. Analytical equations are derived for dynamic slope, dynamic horizontal displacements, and dynamic horizontal strain. Site-specific empirical parameters were derived, which differ from the regional defaults proposed in the literature. Site-specific empirical parameters pertain to both final and dynamic calculations. The model effectively predicts the transitions of final and dynamic horizontal displacement.

Abstract This paper presents an experimental investigation on ultrasonic vibration–assisted magnetorheological finishing (VAMRF) process for improved material removal rate (MRR) and surface finishing on glass optics polishing. An additional process parameter, i.e., vibrating motion, is added in the magnetorheological finishing (MRF) process for corrective polishing of glass optics. Influence function, a material removal characteristic of the process and necessary for deterministic processing, was calculated experimentally for the VAMRF. The results show that hybrid VAMRF provides approximately 20% higher MRR (14.3 nm/min) as compared to that of conventional Ball End MRF (BEMRF), which is 11.9 nm/min. Better surface micro-roughness improvement observed in VAMRF process (3.05 nm) as compared to that in BEMRF process (5.1 nm) from initial value (7.06 nm) in surface of N-BK7 glass workpiece. The developed hybrid process is applied in corrective polishing of glass optics of 25 mm diameter, and it is demonstrated that the figure error (RMS) of the surface has reduced down to 34 nm from 131 nm. Experimental results show that the developed hybrid finishing process is a promising candidate for corrective polishing of optics.

Despite the risk of misspecification they are tied to, parametric models continue to be used in statistical practice because they are simple and convenient to use. In particular, efficient estimation procedures in parametric models are easy to describe and implement. Unfortunately, the same cannot be said of semiparametric and nonparametric models. While the latter often reflect the level of available scientific knowledge more appropriately, performing efficient inference in these models is generally challenging. The efficient influence function is a key analytic object from which the construction of asymptotically efficient estimators can potentially be streamlined. However, the theoretical derivation of the efficient influence function requires specialized knowledge and is often a difficult task, even for experts. In this article, we present a novel representation of the efficient influence function and describe a numerical procedure for approximating its evaluation. The approach generalizes the nonparametric procedures of Frangakis et al. and Luedtke, Carone, and van der Laan to arbitrary models. We present theoretical results to support our proposal and illustrate the method in the context of several semiparametric problems. The proposed approach is an important step toward automating efficient estimation in general statistical models, thereby rendering more accessible the use of realistic models in statistical analyses. Supplementary materials for this article are available online.

Geospatial health risk signals, characterized by associations with high magnitude statistical significance, may frequently originate from circumscribed observational data streams. When these signals are fueled by massive N-size datasets, the large dimensional scale of the sample can induce a misleading interpretation of local evidence as a statistically significant risk inflation. The objective of this study is to verify whether such health risk configurations constitute geospatial structural artifacts: namely, stochastic distortions generated by the spatial information of local health repositories that, despite their massive scale, may remain fundamentally distant from broader contextual realities. To this aim, we present a mathematical framework designed to evaluate geospatial health systemic resilience by resolving local signal inflation through a structural remodelling procedure. By integrating a Recentered Influence Function (RIF) regression and a state-tuple formalization, the model expands the analytical context beyond the boundaries of the local health data, systematically hooking into broader reference frameworks (e.g., historical metrics and national-level standards). Through the application of a spatial Difference-in-Differences (DiD) decomposition, the local risk signal is partitioned into an explained component, justified by local covariates, and a structural unexplained component. This decomposition isolates the divergence between the observed phenomena and a broader baseline, revealing how the magnitude of a local health alarm can often be a function of a measurement bias internal to the data-generating process. To prove the framework’s efficacy, we conducted our analysis on a large geospatial health dataset. The analysis revealed that the initial reported alarm was the byproduct of a 32.5% structural weight deficit in the high-risk stratum of the local experimental population and, most critically, a 45.1% deficit in disease occurrences within the non-exposed local baseline compared to the national reference. This exemplar modeling has demonstrated that our proposed diagnostic and corrective framework can be a useful diagnostic tool to validate the identification of a health risk, having quantified the inversion of the original signal from an initial risk factor of 1.27 to a recalibrated 0.77 value. By isolating the structural difference between local observations and extra-local references, this methodology ensures consistency between verifiable health reality and dataset-specific outcomes, detecting and mitigating structurally inflated risk signals.

Nonparametric correlation estimators as the Kendall and Spearman correlation are widely used in the applied sciences. They are often said to be robust, in the sense of being resistant to outlying observations. In this paper we formally study their robustness by means of their influence functions and gross-error sensitivities. Since robustness of an estimator often comes at the price of an increased variance, we also compute statistical efficiencies at the normal model. We conclude that both the Spearman and Kendall correlation estimators combine a bounded and smooth influence function with a high efficiency. In a simulation experiment we compare these nonparametric estimators with correlations based on a robust covariance matrix estimator.

The industrial robot bonnet polishing platform can not only meet the requirements of high efficiency and precision of optical polishing, but also reduce the system cost. It is a promising polishing equipment solution. The most practical method to improve BP material removal efficiency is to increase the contact depth. It is generally believed that the profile of the tool influence function (TIF) of BP is Gaussian-like distribution traditionally, but it is found that it actually changes into M-shape obviously with large contact depth. However, the existing reports on the TIF of bonnet polishing are mostly based on the Gaussian like TIF model which cannot accurately describe the M-shaped TIF. Therefore, according to the material characteristics of inflatable rubber bonnet, this paper establishes a new model to explain the changes of pressure distribution caused by large contact depth. Furthermore, a bonnet polishing approach based on dual-mode contact depth TIF is proposed in order to improve the removal efficiency in rough polishing stage and increase the convergence accuracy in fine polishing stage.