Robustness and sensitivity analysis of risk measurement procedures Rama Cont, Romain Deguest, Giacomo Scandolo To cite this version: Rama Cont, Romain Deguest, Giacomo Scandolo. {\displaystyle T} ( The level and the power breakdown points of tests are investigated in He, Simpson & Portnoy (1990). … {\displaystyle G} X We introduce three new robustness benchmarks consisting of naturally occurring distribution changes in image style, geographic location, camera operation, and more. By denition, data analysis techniques aim at practical problems of data processing. and i 4:34 Importance of robustness analyses illustrated using Global MPI data. ( ) We choose Robust statistical methods, of which the trimmed mean is a simple example, seek to outperform classical statistical methods in the presence of outliers, or, more generally, when underlying parametric assumptions are not quite correct. 11/20 x ( Strictly speaking, a robust statistic is resistant to errors in the results, produced by deviations from assumptions[1] (e.g., of normality). y x This chapter will deal solely with the topic of robust regression. ; F is defined by: What this actually means is that we are replacing the i-th value in the sample by an arbitrary value and looking at the output of the estimator. ) {\displaystyle \rho } Σ The distribution of standard deviation is erratic and wide, a result of the outliers. T ( The analysis was performed in R and 10,000 bootstrap samples were used for each of the raw and trimmed means. What we are now trying to do is to see what happens to an estimator when we change the distribution of the data slightly: it assumes a distribution, and measures sensitivity to change in this distribution. Note also that robustness analysis is used in other areas of science with yet a different meaning: e.g, in the study of complex systems, robustness analysis is a method of quantifying the effect of uncertainty at the level of the parameters on the final predictions; in statistics, robust estimators are those unaffected by outliers in the data. F function is not critical to gaining a good robust estimate, and many choices will give similar results that offer great improvements, in terms of efficiency and bias, over classical estimates in the presence of outliers.[7]. ) ( {\displaystyle F} The plots below show the bootstrap distributions of the standard deviation, the median absolute deviation (MAD) and the RousseeuwâCroux (Qn) estimator of scale. I feel the term “Robustness” is used quite vaguely in applied statistics. ‖ Σ ; = Given that these conditions of a study are met, the models can be verified to be true through the use of mathematical proofs. Robust Regression Introduction Multiple regression analysis is documented in Chapter 305 – Multiple Regression, so that information will not be repeated here. To this end Ting, Theodorou & Schaal (2007) have recently shown that a modification of Masreliez's theorem can deal with outliers. and solving Properties of an influence function which bestow it with desirable performance are: ρ F θ The data sets for that book can be found via the Classic data sets page, and the book's website contains more information on the data. This may sound a bit ambiguous, but that is because robustness can refer to different kinds of insensitivities to changes. ρ The result is that the modest outlier looks relatively normal. The Kohonen self organising map (KSOM) offers a simple and robust multivariate model for data analysis, thus providing good possibilities to estimate missing values, taking into account its relationship or correlation with other pertinent variables in the data record.[10]. T This example uses: Robust Control Toolbox; Simulink; Open Script. ψ , where arbitrarily large observations) an estimator can handle before giving an incorrect (e.g., arbitrarily large) result. {\displaystyle \lambda ^{*}(T;F):=\sup _{(x,y)\in {\mathcal {X}}^{2} \atop x\neq y}\left\|{\frac {IF(y;T;F)-IF(x;T;F)}{y-x}}\right\|}. What we try to do with MLE's is to maximize {\displaystyle \prod _{i=1}^{n}f(x_{i})} F {\displaystyle x_{1},\dots ,x_{n}} {\displaystyle A} the values {2,3,5,6,9}, then if we add another datapoint with value -1000 or +1000 to the data, the resulting mean will be very different to the mean of the original data. x {\displaystyle F} Maronna, Martin & Yohai (2006) recommend the biweight function with efficiency at the normal set to 85%. For an example of robustness, we will consider t-procedures, which include the confidence interval for a population mean with unknown population standard deviation as well as hypothesis tests about the population mean. | {\displaystyle \nu =1} Factors affecting robustness. ? Abstract. ( n Intuitively, the breakdown point of an estimator is the proportion of incorrect observations (e.g. See Huber (1981). or, equivalently, minimize ∑ The use of t-procedures assumes the following: In practice with real-life examples, statisticians rarely have a population that is normally distributed, so the question instead becomes, “How robust are our t-procedures?”. 1 2. Theoretically, Quantitative Finance, Taylor & Francis (Routledge), 2010, 10 (6), pp.593 - 606. In principle, . Robustness to distributional assumptions is an important consideration throughout statistics, so it is important to emphasize that quantile regression inherits robustness properties of the ordinary sample quantiles. ) ; The accuracy of the estimate depends on how good and representative the model is and how long the period of missing values extends. = . M-estimators are a generalization of maximum likelihood estimators (MLEs). , (The mathematical context of this paragraph is given in the section on empirical influence functions.). ) For the t-distribution with , := n These considerations do not "invalidate" M-estimation in any way. in ρ {\displaystyle \gamma ^{*}(T;F):=\sup _{x\in {\mathcal {X}}}|IF(x;T;F)|}, λ d [2] The plots are based on 10,000 bootstrap samples for each estimator, with some Gaussian noise added to the resampled data (smoothed bootstrap). > In Use Case Driven Object Modeling With UML Doug Rosenberg and Kendall Scott describe a technique called robustness analysis. ∈ The problem is even worse in higher dimensions. ; , That a statistical analysis is not robust with respect to the framing of the model should mean roughly that small changes in the inputs cause large changes in the outputs. f This example shows how to use Simulink® blocks and helper functions provided by Robust Control Toolbox™ to specify and analyze uncertain systems in Simulink and how to use these tools to perform Monte Carlo simulations of uncertain systems. , In fact, the mean, median and trimmed mean are all special cases of M-estimators. n i ρ F ) ) The outliers in the speed-of-light data have more than just an adverse effect on the mean; the usual estimate of scale is the standard deviation, and this quantity is even more badly affected by outliers because the squares of the deviations from the mean go into the calculation, so the outliers' effects are exacerbated. Robust methods provide automatic ways of detecting, downweighting (or removing), and flagging outliers, largely removing the need for manual screening. ( lim ) M-estimators do not necessarily relate to a probability density function. | We're looking at: Therefore, manual screening for outliers is often impractical. {\displaystyle G=\Delta _{x}} Full Robustness in Bayesian Modelling of a Scale Parameter Desgagné, Alain, Bayesian Analysis, 2013 Data-adaptive trimming of the Hill estimator and detection of outliers in the extremes of heavy-tailed data Bhattacharya, Shrijita, Kallitsis, Michael, and Stoev, Stilian, Electronic Journal of Statistics, 2019 Traditionally, statisticians would manually screen data for outliers, and remove them, usually checking the source of the data to see whether the outliers were erroneously recorded. ρ Of the 60 quantitative articles published in 2010, the vast majority - 85 percent - contained at least one footnote referencing an unreported analysis purporting to confirm the robustness of the main results (see Table 1). The mean is not a robust measure of central tendency. Stability Robustness Expand/collapse global location ... we present a few examples to illustrate the use of the small-gain theorem in stability robustness analysis. ; : I Based on the nonlinear gap metric robustness analysis, a study is undertaken for nonlinear systems with input-output linearizing controllers to derive and validate theoretical robustness and performance margins for these systems. {\displaystyle \psi } robustness: ( rō-bust'ness ), In statistics, the degree to which the probability of drawing a wrong conclusion from the test result is not seriously affected by moderate departures from the assumptions implicit in the model on which the test is based. ( x We find that using larger models and synthetic data augmentation can improve robustness … {\displaystyle x_{1},\dots ,x_{n}} x Using our benchmarks, we take stock of previously proposed hypotheses for out-of-distribution robustness and put them to the test. x If we replace one of the values with a datapoint of value -1000 or +1000 then the resulting median will still be similar to the median of the original data. Robustness refers to “the sensitivity of the overall conclusions to various limitations of the data, assumptions, and analytic approaches to data analysis” . is a sample from these variables. {\displaystyle t} , The median is a robust measure of central tendency. y Robust statistics is about developing procedures with levels of performance that are consistently high for processes that obey realistic deviations from the model, i.e.

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