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AbstractIn contrast to using means, there are two distinct approaches to developing robust measures effect size when dealing with two dependent variables. One is based on difference scores and the other focuses on the marginal distributions. The former approach is trivial based on extant results. This paper focuses on new robust methods based on the marginal distributions that take into account the strength of the association between the two dependent variables. Simulations are used to assess the extent a reasonably accurate confidence interval can be computed. Included is a proposed method for situations where there is a covariate. [ABSTRACT FROM AUTHOR]

We consider the multiple change point problem in a general framework based on estimating equations. This extends classical sample mean-based methodology to include robust methods but also different types of changes such as changes in linear regression or changes in count data including Poisson autoregressive time series. In this framework, we derive a general theory proving consistency for the number of change points and rates of convergence for the estimators of the locations of the change points. More precisely, two different types of MOSUM (moving sum) statistics are considered: A MOSUM-Wald statistic based on differences of local estimators and a MOSUM-score statistic based on a global inspection parameter. The latter is usually computationally less involved in particular in nonlinear problems where no closed form of the estimator is known such that numerical methods are required. Finally, we evaluate the methodology by some simulations as well as using geophysical well-log data. [ABSTRACT FROM AUTHOR]

The work develops the theory of stable M-estimators belonging to the class of redescending estimators, having the property of resistance to asymmetric contamination. Many well-known redescending estimators can be obtained within the framework of the locally stable approach of A.M. Shurygin, based on the analysis of the estimator instability functional (L2-norm of the influence function), or his approach based on the model of a series of samples with random point contamination (point Bayesian contamination model). These approaches are convenient for constructing various stable M-estimators and, in comparison with classical robust procedures, provide wider opportunities. The family of conditionally optimal estimators proposed by A.M. Shurygin within the framework of the first of the listed approaches can be defined as optimizing the asymptotic dispersion under a constraint on the value of instability. The corresponding problem can be represented in the form of optimization of the weighted L2-norm of the influence function. The second approach considers a specially formed nonparametric neighborhood of the model distribution, and it can also be reduced to the analysis of the weighted L2-norm of the influence function. Thus, this estimation quality criterion is quite general and useful for constructing robust estimators. The theory of estimators that are optimal in terms of weighted L2-norm of the influence function is currently underdeveloped. Specifically, for the corresponding families of estimators, the question of the uniqueness of family members remains unresolved. The question comes down to studying the convexity (concavity) of the optimized functional depending on the parameter defining the family. In the presented work, an expression is obtained in general form for the derivative with respect to the parameter of the quality functional of the optimal estimator. Inequalities are obtained for the second derivative necessary to establish its convexity (concavity) with respect to the parameter. Corollaries from these results are applied to describe the properties of a conditionally optimal family. The influence functions of a number of conditionally optimal estimators for the shift and scale parameters of the normal model are constructed. The characteristics of these estimators are studied. The stability of most of the considered estimators is shown, which is important for their practical application. The theoretical results obtained can be useful in studying the properties of compromise estimators based on two criteria as well as in studying minimax contamination levels within the framework of A.M. Shurygin’s point Bayesian contamination model. The results of the work can be used in situations of purposed data corruption by an adversary including the problems related to adversarial machine learning.

It is shown that higher-order influence functions for M-estimators are mathematically equivalent to higher-order stochastic approximations to these estimators. The stochastic expansions are also shown to have corresponding higher-order U-statistic representations, providing an alternative approach for deriving and analyzing the approximate properties of M-estimators. [ABSTRACT FROM AUTHOR]

A generic out-of-sample error estimate is proposed for |$M$| -estimators regularized with a convex penalty in high-dimensional linear regression where |$(\boldsymbol{X},\boldsymbol{y})$| is observed and the dimension |$p$| and sample size |$n$| are of the same order. The out-of-sample error estimate enjoys a relative error of order |$n^{-1/2}$| in a linear model with Gaussian covariates and independent noise, either non-asymptotically when |$p/n\le \gamma $| or asymptotically in the high-dimensional asymptotic regime |$p/n\to \gamma ^{\prime}\in (0,\infty)$|⁠. General differentiable loss functions |$\rho $| are allowed provided that the derivative of the loss is 1-Lipschitz; this includes the least-squares loss as well as robust losses such as the Huber loss and its smoothed versions. The validity of the out-of-sample error estimate holds either under a strong convexity assumption, or for the L1-penalized Huber M-estimator and the Lasso under a sparsity assumption and a bound on the number of contaminated observations. For the square loss and in the absence of corruption in the response, the results additionally yield |$n^{-1/2}$| -consistent estimates of the noise variance and of the generalization error. This generalizes, to arbitrary convex penalty and arbitrary covariance, estimates that were previously known for the Lasso. [ABSTRACT FROM AUTHOR]

Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M‐type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are subject to measurement error. We show that under suitable assumptions the proposed family of estimators is minimax rate optimal both for commonly and independently observed trajectories and we illustrate its highly competitive performance and practical usefulness in a Monte‐Carlo study and a real‐data example involving recent Covid‐19 data. [ABSTRACT FROM AUTHOR]

M-type smoothing splines are a broad class of spline estimators that include the popular least-squares smoothing spline but also spline estimators that are less susceptible to outlying observations and model misspecification. However, available asymptotic theory only covers smoothing spline estimators based on smooth objective functions and consequently leaves out frequently used resistant estimators such as quantile and Huber-type smoothing splines. We provide a general treatment in this paper and, assuming only the convexity of the objective function, show that the least-squares (super-)convergence rates can be extended to M-type estimators whose asymptotic properties have not been hitherto described. We further show that auxiliary scale estimates may be handled under significantly weaker assumptions than those found in the literature and we establish optimal rates of convergence for the derivatives, which have not been obtained outside the least-squares framework. A simulation study and a real-data example illustrate the competitive performance of non-smooth M-type splines in relation to the least-squares spline on regular data and their superior performance on data that contain anomalies. [ABSTRACT FROM AUTHOR]

A confidence distribution is a distribution for a parameter of interest based on a parametric statistical model. As such, it serves the same purpose for frequentist statisticians as a posterior distribution for Bayesians, since it allows to reach point estimates, to assess their precision, to set up tests along with measures of evidence, to derive confidence intervals, comparing the parameter of interest with other parameters from other studies, etc. A general recipe for deriving confidence distributions is based on classical pivotal quantities and their exact or approximate distributions. However, in the presence of model misspecifications or outlying values in the observed data, classical pivotal quantities, and thus confidence distributions, may be inaccurate. The aim of this paper is to discuss the derivation and application of robust confidence distributions. In particular, we discuss a general approach based on the Tsallis scoring rule in order to compute a robust confidence distribution. Examples and simulation results are discussed for some problems often encountered in practice, such as the two-sample heteroschedastic comparison, the receiver operating characteristic curves and regression models. [ABSTRACT FROM AUTHOR]

Performance of any control scheme in Phase II depends directly on the quality of estimators utilized in Phase I. In practice, outliers could be present in the data which would impact the performance of estimators adversely. This study deals with robust parameter estimation and monitoring linear profiles in the presence of outliers and compares the results with the least squares (LS) estimators. For this purpose, M-estimators are used as robust estimators and empirical distributions for related statistics are determined using Mont Carlo simulation to calculate control limits for two T 2 control charts and for codding independent variable method. Using a numerical example, profile parameters are estimated by ordinary least squares and M-estimators and the resulting statistics are monitored by two T 2 control schemes. Phase II control charts are determined based on the two types of estimators and compared for different out of control profiles. Empirical distributions did not follow their exact distributions obtained by least squares method. Simulation results confirm that M-estimators lead to better estimates in comparison to LS estimators and also improves classification performance. Robust estimators also lead to improvement in ARL performance in comparison to LS estimators. [ABSTRACT FROM AUTHOR]

Ridge regression is used to circumvent the problem of multicollinearity in the multiple linear regression models. Beside the multicollinearity, when the outliers in the y-direction are also present, then the usual ridge regression estimators gives inefficient results in terms of mean squared error (MSE). In order to mitigate such situation, ridge M-estimators are often used. Several estimators are available in literature but they do not perform well in terms of MSE when the joint problem of high multicollinearity and y-direction outliers is present. In this article, some new quantile based ridge M-estimators are proposed. The new estimators are then compared with other existing estimators through extensive Monte Carlo simulations for various error term distributions, degrees of multicollinearity and percentage of y-direction outliers. Based on simulation study with minimum MSE criterion, the new estimators outperform in many considered scenarios. Particularly, in case of high multicollinearity, y-direction outliers and heavy tailed error distributions, the proposed estimators have shown efficient results. A numerical example is also presented to support the simulation results. [ABSTRACT FROM AUTHOR]

Outliers in a statistical analysis strongly affect the performance of the ordinary least squares, such outliers need to be detected and extreme outliers deleted. This paper is aimed at proposing a redescending M-estimator, which is more efficient and robust, compared to other existing redescending M-estimators. The proposed method is applied to real life data to verify its effectiveness in detecting and deleting of outliers. The Monte Carlo simulation method is also used to investigate the performance of the newly proposed method. The results from the real life data and the Monte Carlo simulation method show that the proposed method is effective in the detection and deletion of extreme outliers compared to other existing redescending M-estimators. [ABSTRACT FROM AUTHOR]

Recently, 5G networks have emerged as a new technology that can control the advancement of telecommunication networks and transportation systems. Furthermore, 5G networks provide better network performance while reducing network traffic and complexity compared to current networks. Machine-learning techniques (ML) will help symmetric IoT applications become a significant new data source in the future. Symmetry is a widely studied pattern in various research areas, especially in wireless network traffic. The study of symmetric and asymmetric faults and outliers (anomalies) in network traffic is an important topic. Nowadays, deep learning (DL) is an advanced approach in challenging wireless networks such as network management and optimization, anomaly detection, predictive analysis, lifetime value prediction, etc. However, its performance depends on the efficiency of training samples. DL is designed to work with large datasets and uses complex algorithms to train the model. The occurrence of outliers in the raw data reduces the reliability of the training models. In this paper, the performance of Vehicle-to-Everything (V2X) traffic was estimated using the DL algorithm. A set of robust statistical estimators, called M-estimators, have been proposed as robust loss functions as an alternative to the traditional MSE loss function, to improve the training process and robustize DL in the presence of outliers. We demonstrate their robustness in the presence of outliers on V2X traffic datasets.

Recently, 5G networks have emerged as a new technology that can control the advancement of telecommunication networks and transportation systems. Furthermore, 5G networks provide better network performance while reducing network traffic and complexity compared to current networks. Machine-learning techniques (ML) will help symmetric IoT applications become a significant new data source in the future. Symmetry is a widely studied pattern in various research areas, especially in wireless network traffic. The study of symmetric and asymmetric faults and outliers (anomalies) in network traffic is an important topic. Nowadays, deep learning (DL) is an advanced approach in challenging wireless networks such as network management and optimization, anomaly detection, predictive analysis, lifetime value prediction, etc. However, its performance depends on the efficiency of training samples. DL is designed to work with large datasets and uses complex algorithms to train the model. The occurrence of outliers in the raw data reduces the reliability of the training models. In this paper, the performance of Vehicle-to-Everything (V2X) traffic was estimated using the DL algorithm. A set of robust statistical estimators, called M-estimators, have been proposed as robust loss functions as an alternative to the traditional MSE loss function, to improve the training process and robustize DL in the presence of outliers. We demonstrate their robustness in the presence of outliers on V2X traffic datasets. [ABSTRACT FROM AUTHOR]