In experimental science measurements are typically repeated only a few times, yielding a sample size n of the order of 3 to 8. One then wants to summarize the measurements by a central value and measure their variability, i.e. estimate location and scale. These estimates should preferably be robust against outliers, as reflected by their small-sample breakdown value. The estimator's stylized empirical influence function should be smooth, monotone increasing for location, and decreasing–increasing for scale. It turns out that location can be estimated robustly for n⩾3, whereas for scale n⩾4 is needed. Several well-known robust estimators are studied for small n, yielding some surprising results. For instance, the Hodges–Lehmann estimator equals the average when n=4. Also location M-estimators with auxiliary scale are studied, addressing issues like the difference between one-step and fully iterated M-estimators. Simultaneous M-estimators of location and scale (‘Huber's Proposal 2’) are considered as well, and it turns out that their lack of robustness is already noticeable for such small samples. Recommendations are given as to which estimators to use.
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... We extend the robust mean estimator [13,29] to the context of QOI estimation. Specifically, given the set of queries X (|X| = n query ), we estimate the QOI in an iterative manner, in which the k-th round estimatorx (k) is updated as follows: ...
... The value ofx (0) is initialized as the median of the query set X. As observed in our empirical evaluation (which is also consistent with the results of [29]), one-iteration estimationx (1) is typically accurate enough to approximate the QOI x. Thus we set the number of iterations k = 1 by default. ...
... We extend the robust mean estimator [13,30] to the context of QOI estimation. Specifically, given the set of queries X (|X| = n query ), we estimate the QOI in an iterative manner, in which the k-th round estimatorx (k) is updated as follows: ...
... The value ofx (0) is initialized as the median of the query set X. As observed in our empirical evaluation (which is also consistent with the results of [30]), one-iteration estimationx (1) is typically accurate enough to approximate the QOI x. Thus we set the number of iterations k = 1 by default. ...
Deep neural networks (DNNs) are inherently susceptible to adversarial attacks even under black-box settings, in which the adversary only has query access to the target models. In practice, while it may be possible to effectively detect such attacks (e.g., observing massive similar but non-identical queries), it is often challenging to exactly infer the adversary intent (e.g., the target class of the adversarial example the adversary attempts to craft) especially during early stages of the attacks, which is crucial for performing effective deterrence and remediation of the threats in many scenarios. In this paper, we present AdvMind, a new class of estimation models that infer the adversary intent of black-box adversarial attacks in a robust and prompt manner. Specifically, to achieve robust detection, AdvMind accounts for the adversary adaptiveness such that her attempt to conceal the target will significantly increase the attack cost (e.g., in terms of the number of queries); to achieve prompt detection, AdvMind proactively synthesizes plausible query results to solicit subsequent queries from the adversary that maximally expose her intent. Through extensive empirical evaluation on benchmark datasets and state-of-the-art black-box attacks, we demonstrate that on average AdvMind detects the adversary intent with over 75% accuracy after observing less than 3 query batches and meanwhile increases the cost of adaptive attacks by over 60%. We further discuss the possible synergy between AdvMind and other defense methods against black-box adversarial attacks, pointing to several promising research directions.
... To solve the equation iteratively, usually the MAD is taken as initial scale estimate. Also simultaneous M-estimators of location and scale can be considered, but they have a smaller breakdown value, even in small samples [10]. ...
In this article, we review the most important robust statistical methods for low-dimensional data which have far more observations than variables. The focus is on methods that are used in comprehensive chemometrics. We first review robust estimators of location and scale, and show how to adjust boxplots when dealing with skewed distributions. Then robust estimators for covariance and their use in multivariate analysis are discussed. Also robust correlation and robust regression methods are reviewed. We also provide a software list.
... The RTs were log transformed (natural base) to improve normality. Moreover, to obtain central tendency measures that were as robust as possible against the influence of outliers, a robust M-estimator of location (robust mean RTs; for details, see ; see also Rousseeuw & Verboven, 2002) was computed for each participant and condition separately using the LIBRA MATLAB library (Verboven & Hubert, 2005, 2010. After these transformations, the variables showed acceptable skewness and kurtosis (all < |0.63|). ...
Intrinsic brain dynamics may play an important role in explaining inter-individual variability in executive functions. In the present electroencephalography (EEG) study, we focused on the brain lateralization patterns predicting performance on three different monitoring tasks of temporal, verbal and spatial nature. These tasks were administered to healthy young participants after their EEG was recorded during a resting state session. Behavioral indices of monitoring efficiency were computed for each task and a source-based spectral analysis was performed on participants' resting-state EEG activity. A lateralization index was then computed for each of 75 homologous cortical regions as the right-left difference score for the log-transformed power ratio between beta and alpha frequencies. Finally, skipped Pearson correlations between the lateralization index in each cortical region and behavioral performance of the three monitoring tasks were computed. An intersection among the three tasks showed that right-lateralization in different prefrontal regions, including the middle frontal gyrus, was positively correlated with monitoring abilities across the three tasks. In conclusion, right-lateralized brain mechanisms set the stage for the ability to monitor for targets in the environment, independently of the specific task characteristics. These mechanisms are grounded in hemispheric asymmetry dynamics already observable at rest.
... The arithmetic mean, also called average, and the standard deviation have a breakdown of 0% and are therefore not good estimators for small dataset and even less for non-Gaussian and nonsymmetric, distributions. Using them in such conditions would lead to significantly biased estimations [10]. ...
... The radir R package and Bayesian methods were applied for dose and irradiated fraction estimation (Moriña et al. 2015;Higueras et al. 2015Higueras et al. , 2016. Robust algorithms recommended for very small samples were applied to obtain the mean and the standard deviation (SD) from the four labs (Rousseeuw and Verboven 2002;Hampel et al. 2011;ISO 2015). The mean and SD of the dose the CV and trueness were calculated as previously reported (DiGiorgio et al. 2011;Garc ıa et al. 2013). ...
Purpose: This paper describes how the BioDoseNet Image Repository was used as a training tool for the dicentric assay.
Materials and methods: The training was implemented in three phases, introduction to dicentric scoring, dose response curve elaboration and dose assessment exercise. Four labs without previous experience in the dicentric assay participated and four modules of the Repository were used.
Results: The labs become familiar with aberrations induced by ionizing radiation. The labs were able to generate data for the elaboration of a dose response curve and then successfully estimated doses and irradiated fractions in 6 blind samples.
Conclusion: The performance of these laboratories during the exercise demonstrates the efficacy of the BioDoseNet Image Repository as a training tool and the utility of web based scoring for the dicentric assay community.
... Next, in order to obtain central tendency measures of participants' performance that were as robust as possible against the influence of outliers, we computed for each participant and monitoring condition a robust M-estimator of location (robust mean RTs, see Ambrosini & Vallesi, 2016). This estimation procedure uses the logistic psi-function and the median absolute deviation as the auxiliary scale estimate, as implemented by the mloclogist and madc functions in the LIBRA Matlab library (Verboven & Hubert, 2005, and it is robust to non-normality and sample size (Rousseeuw & Verboven, 2002). From the difference between the performance in the nonmonitoring blocks and that for non-target trials of the monitoring blocks, we estimated the monitoring effects of the spatial and verbal tasks, which are thought to depend on the activity of right-lateralized fronto-parietal regions mediating monitoring processes (Vallesi, 2012; see also Capizzi et al., 2016b). ...
Executive functions refer to high-level cognitive processes that, by operating on lower-level mental processes, flexibly regulate and control our thoughts and goal-directed behavior. Despite their crucial role, the study of the nature and organization of executive functions still faces inherent difficulties. Moreover, most executive function models put under test until now are brain-free models: they are defined and discussed without assumptions regarding the neural bases of executive functions. By using a latent variable approach, here we tested a brain-centered model of executive function organization proposing that two distinct domain-general executive functions, namely, criterion setting and monitoring, may be dissociable both functionally and anatomically, with a left vs. right hemispheric preference of prefrontal cortex and related neural networks, respectively. To this end, we tested a sample of healthy participants on a battery of computerized tasks assessing criterion setting and monitoring processes and involving diverse task domains, including the verbal and visuospatial ones, which are well-known to be lateralized. By doing this, we were able to specifically assess the influence of these task domains on the organization of executive functions and to directly contrast a process-based model of EF organization versus both a purely domain-based model and a process-based, but domain-dependent one. The results of confirmatory factor analyses showed that a purely process-based model reliably provided a better fit to the observed data as compared to alternative models, supporting the specific theoretical model that fractionates a subset of executive functions into criterion setting and monitoring with hemispheric specializations emerging regardless of the task domain.
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... In Equations 6 and 7, í µí± " and í µí± " are the correction factors which make MAD n and QN unbiased estimators respectively. While í µí± " has also 50% breakdown point and bounded influence function, but, discontinuities in the influence functions of MAD n and QN make the application of these estimators less favorable in small samples ( Rousseuw and Verboven, 2002). The advantage QN over MAD n is its high Gaussian efficiency (≈83 %). ...
... In Equations 6 and 7, " and " are the correction factors which make MAD n and QN unbiased estimators respectively. While " has also 50% breakdown point and bounded influence function, but, discontinuities in the influence functions of MAD n and QN make the application of these estimators less favorable in small samples (Rousseuw and Verboven, 2002). The advantage QN over MAD n is its high Gaussian efficiency (≈83 %). ...
Maintaining the quality of manufactured products at a desired level is known to increase customer satisfaction and profitability. Shewhart control chart is the most widely used in statistical process control (SPC) technique to monitor the quality of products and control process variability. Based on the assumption of independent and normally distributed data sets, sample mean and standard deviation statistics are known to be the most efficient conventional estimators to determine the process location and scale, respectively. On the other hand, there is not guarantee that the real-world process data would be normally distributed: outliers may exist, and/or sampled population may be contaminated. In such cases, efficiency of the conventional estimators is significantly reduced, and power of the Shewhart charts may be undesirably low, e.g. occasional outliers in the rational subgroups (Phase I dataset) may drastically affect the sample mean and standard deviation, resulting a serious delay in detection of inferior products (Phase II procedure). For more efficient analyses, it is required to use robust estimators against contaminations. Consequently, it is determined that robust estimators are more efficient both against diffuse localized and symmetric-asymmetric contaminations, and have higher power in detecting disturbances, compared to conventional methods.
This paper concerns squared Msplit(q) estimation and its robustness against outliers. Previous studies in this field have been based on theoretical approaches. It has been proven that a conventional analysis of robustness is insufficient for Msplit(q) estimation. This is due to the split of the functional model into q competitive ones and, hence, the estimation of q competitive versions of the parameters of such models. Thus, we should consider robustness from the global point of view (traditional approach) and from the local point of view (robustness in relation between two “neighboring” estimates of the parameters). Theoretical considerations have generally produced many interesting findings about the robustness of Msplit(q) estimation and the robustness of the squared Msplit(q) estimation, although some of features are asymptotic. Therefore, this paper is focused on empirical analysis of the robustness of the squared Msplit(q) estimation for finite samples and, hence, it produces information on robustness from a more practical point of view. Mostly, the analyses are based on Monte Carlo simulations. A different number of observation aggregations are considered to determine how the assumption of different values of q influence the estimation results. The analysis shows that local robustness (empirical local breakdown points) is fully compatible with the theoretical derivations. Global robustness is highly dependent on the correct assumption regarding q. If it suits reality, i.e. if we predict the number of observation aggregations and the number of outliers correctly, then the squared Msplit(q) estimation can be an alternative to classical robust estimations. This is confirmed by empirical comparisons between the method in question and the robust M-estimation (the Huber method). On the other hand, if the assumed value of q is incorrect, then the squared Msplit(q) estimation usually breaks down.
We consider a new class of scale estimators with 50% breakdown point. The estimators are defined as order statistics of certain subranges. They all have a finite-sample breakdown point of [n/2]/n, which is the best possible value. (Here, [...] denotes the integer part.) One estimator in this class has the same influence function as the median absolute deviation and the least median of squares (LMS) scale estimator (i.e., the length of the shortest half), but its finite-sample efficiency is higher. If we consider the standard deviation of a subsample instead of its range, we obtain a different class of 50% breakdown estimators. This class contains the least trimmed squares (LTS) scale estimator. Simulation shows that the LTS scale estimator is nearly unbiased, so it does not need a small-sample correction factor. Surprisingly, the efficiency of the LTS scale estimator is less than that of the LMS scale estimator.
This paper reviews some aspects of positive-breakdown regression that have been discussed. Apart from efficiency, also some related topics are addressed in order to obtain a broader view. Several unusual aspects are shown to be intimately connected with the exact fit property. It is argued that the latter is not a drawback but an interesting property, which helps to explain why positive-breakdown estimators often succeed at revealing a hidden structure in the data.
In robust estimation one frequently needs an initial or auxiliary estimate of scale. For this one usually takes the median absolute deviation MADn = 1.4826 med, {|xi − medj x j |}, because it has a simple explicit formula, needs little computation time, and is very robust as witnessed by its bounded influence function and its 50% breakdown point. But there is still room for improvement in two areas: the fact that MADn is aimed at symmetric distributions and its low (37%) Gaussian efficiency. In this article we set out to construct explicit and 50% breakdown scale estimators that are more efficient. We consider the estimator Sn = 1.1926 med, {medj | xi − xj|} and the estimator Qn given by the .25 quantile of the distances {|xi − x j |; i < j}. Note that Sn and Qn do not need any location estimate. Both Sn and Qn can be computed using O(n log n) time and O(n) storage. The Gaussian efficiency of Sn is 58%, whereas Qn attains 82%. We study Sn and Qn by means of their influence functions, their bias curves (for implosion as well as explosion), and their finite-sample performance. Their behavior is also compared at non-Gaussian models, including the negative exponential model where Sn has a lower gross-error sensitivity than the MAD.
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