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In semiparametric and nonparametric statistical inference, the asymptotic normality of estimators has been widely established when they are √n-consistent. In many applications, nonparametric estimators are not able to achieve this rate. We have a result on the asymptotic normality of nonparametric M-estimators that can be used if the rate of convergence of an estimator is n-1/2 or slower. We apply this to study the asymptotic distribution of sieve estimators of functionals of a mean function from a counting process, and develop nonparametric tests for the problem of treatment comparison with panel count data. The test statistics are constructed with spline likelihood estimators instead of nonparametric likelihood estimators. The new tests have a more general and simpler structure and are easy to implement. Simulation studies show that the proposed tests perform well even for small sample sizes. We find that a new test is always powerful for all the situations considered and is thus robust. For illustration, a data analysis example is provided. [ABSTRACT FROM AUTHOR]

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To perform the image segmentation task, in this Letter, a kernel fuzzy C‐means algorithm is introduced, strengthened by a robust Gaussian radial basis function kernel based on M‐estimators. It is well‐known that these kernels consider the squared difference as a similarity measure, which is not robust to atypical data. In this regard, the main motivation of this contribution is to improve the atypical information tolerance of these kernels, in order to make a better clustering of pixels. Experimental tests were developed considering colour images. The robustness and effectiveness of this proposal are verified by quantitative and qualitative results. [ABSTRACT FROM AUTHOR]

The Aumann-type mean has been shown to possess valuable properties as a measure of the location or central tendency of fuzzy data associated with a random experiment. However, concerning robustness its behavior is not appropriate. The Aumann-type mean is highly affected by slight changes in the fuzzy data or when outliers arise in the sample. Robust estimators of location, on the other hand, avoid such adverse effects. For this purpose, this paper considers the M-estimation approach and discusses conditions under which this alternative yields valid fuzzy-valued M-estimators. The resulting M-estimators are applied to a real-life example. Finally, some simulation studies show empirically the suitability of the introduced estimators. [ABSTRACT FROM PUBLISHER]

In this article, we propose three M-estimators for multiple regression model when response variable is subject to twice censoring. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to construct interval estimators. [ABSTRACT FROM PUBLISHER]

Classical statistics relies largely on parametric models. Typically, assumptions are made on the structural and the stochastic parts of the model and optimal procedures are derived under these assumptions. Standard examples are least squares estimators in linear models and their extensions, maximum-likelihood estimators and the corresponding likelihood-based tests, and generalized methods of moments (GMM) techniques in econometrics. Robust statistics deals with deviations from the stochastic assumptions and their dangers for classical estimators and tests and develops statistical procedures that are still reliable and reasonably efficient in the presence of such deviations. It can be viewed as a statistical theory dealing with approximate parametric models by providing a reasonable compromise between the rigidity of a strict parametric approach and the potential difficulties of interpretation of a fully nonparametric analysis. Many classical procedures are well known for not being robust. These procedures are optimal when the assumed model holds exactly, but they are biased and/or inefficient when small deviations from the model are present. The statistical results obtained from standard classical procedures on real data applications can therefore be misleading. In this paper we will give a brief introduction to robust statistics by reviewing some basic general concepts and tools and by showing how they can be used in data analysis to provide an alternative complementary analysis with additional useful information. In this study, we focus on robust statistical procedures based on M-estimators and tests because they provide a unified statistical framework that complements the classical theory. Robust procedures will be discussed for standard models, including linear models, general linear model, and multivariate analysis. Some recent developments in high-dimensional statistics will also be outlined. WIREs Comput Stat 2015, 7:372-393. doi: 10.1002/wics.1363 For further resources related to this article, please visit the . [ABSTRACT FROM AUTHOR]

In this letter, we propose an alternative proof for the uniqueness of Maronna's M-estimator of scatter for N vector observations y1, ..., yN ∈ Rm under a mild constraint of linear independence of any subset of m of these vectors. This entails in particular almost sure uniqueness for random vectors yi with a density as long as N > m. This approach allows to establish further relations that demonstrate that a properly normalized Tyler's M-estimator of scatter can be considered as a limit of Maronna's M-estimator. More precisely, the contribution is to show that each M-estimator, verifying some mild conditions, converges towards a particular Tyler's M-estimator. These results find important implications in recent works on the large dimensional (random matrix) regime of robust M-estimation. [ABSTRACT FROM AUTHOR]

Unlike with independent data, smoothed bootstraps have received little consideration for time series, although data smoothing within resampling can improve bootstrap approximations, especially when target distributions depend on smooth population quantities (e.g., marginal densities). For approximating a broad class statistics formulated through statistical functionals (e.g., LL-estimators, and sample quantiles), we propose a smooth bootstrap by modifying a state-of-the-art (extended) tapered block bootstrap (TBB). Our treatment shows that the smooth TBB applies to time series inference cases not formally established with other TBB versions. Simulations also indicate that smoothing enhances the block bootstrap. [ABSTRACT FROM AUTHOR]

Most of the work done on self-organizing maps relies on the minimization of the mean squared error. This nonrobust approach leads to poor performance in the presence of outliers. Here we consider robust M-estimators as an alternative for least squares in the context of self-organization. New learning rules are derived, so that the original Kohonen׳s SOFM learning rule is a particular case. Experimental results are presented which demonstrate the robustness of our method against outliers, when compared to other robust self-organizing maps. [ABSTRACT FROM AUTHOR]

Response surfaces express the behavior of responses and can be used for both single and multi-response problems. A common approach to estimate a response surface using experimental results is the ordinary least squares (OLS) method. Since OLS is very sensitive to outliers, some robust approaches have been discussed in the literature. Although there are many methods available in the literature for multiple response optimizations, there are a few studies in model building especially robust models. Assuming correlated responses, in this paper, a robust coefficient estimation method is proposed for multi response problem based on M-estimators. In order to illustrate the performance of the proposed procedure, a contaminated experimental design using a numerical example available in the literature with some modifications is used. Both the classical multivariate least squares method and the proposed robust multivariate approach are used to estimate regression coefficients of multi-response surfaces based on this example. Moreover, a comparison of the proposed robust multi response surface (RMRS) approach with separate robust estimation of single response show that the proposed approach is more efficient. [ABSTRACT FROM AUTHOR]

Abstract: Procedures detecting a change of regression parameters in a linear model are considered when both the regressors and the errors are weakly dependent in the sense of L p –m-approximability. M-estimators and weighted M-residuals are used to construct test statistics, and their asymptotic distribution is studied under the null hypothesis of no change and under contiguous alternatives. Estimators of a long-run variance are also considered and studied both analytically and numerically. [Copyright &y& Elsevier]

Abstract: The introduction of arterial spin labelling (ASL) techniques in magnetic resonance imaging (MRI) has made feasible a non-invasive measurement of the cerebral blood flow (CBF). However, to date, the low signal-to-noise ratio of ASL gives us no option but to repeat the acquisition to accumulate enough data in order to get a reliable signal. The perfusion signal is then usually extracted by averaging across the repetitions. But the sample mean is very sensitive to outliers. A single incorrect observation can therefore be the source of strong detrimental effects on the perfusion-weighted image estimated with the sample mean. We propose to estimate robust ASL CBF maps with M-estimators to overcome the deleterious effects of outliers. The behavior of this method is compared to z-score thresholding as recommended in Tan et al. (Journal of Magnetic Resonance Imaging 2009;29(5):1134–9.). Validation on simulated and real data is provided. Quantitative validation is undertaken by measuring the correlation with the most widespread technique to measure perfusion with MRI: dynamic susceptibility weighted contrast imaging. [Copyright &y& Elsevier]

Abstract: In this paper, we make use of the technique of martingales to establish the moderate deviations and strong Bahadur representations for M-estimators of the regression parameter in a linear model when the errors form a type of Markov chain. As an application, we obtain a law of the iterated logarithm for the estimators. [Copyright &y& Elsevier]

Abstract: In this paper we propose a family of robust estimators for generalized linear models. The basic idea is to use an M-estimator after applying a variance stabilizing transformation to the response. We show the consistency and asymptotic normality of these estimators. We also obtain a lower bound for their breakdown point. A Monte Carlo study shows that the proposed estimators compare favorably with respect to other robust estimators for generalized linear models with Poisson response and log link. [Copyright &y& Elsevier]

In this paper we study bias-corrections to the weighted MLE (Dupuis and Morgenthaler, 2002), a robust estimator simply defined through a weighted score function. Indeed, although the WMLE is relatively simple to compute, for most models it is not consistent and hence not very helpful. For example, the model we consider in this paper is the generalized linear latent variable model (GLLVM) proposed in Moustaki and Knott (2000) (see also Moustaki, 1996, Sammel, Ryan, and Legler, 1997 and Bartholomew and Knott, 1999). The score functions of this model are very complicated. They contain integrals that need to be evaluated. Moreover, they are highly nonlinear in the parameters which makes the use of complicated robust estimator quite impossible in practice. Moustaki and Victoria-Feser (2006) propose to use a weighted MLE and develop indirect inference (Gouri´eroux, Monfort, and Renault, 1993, Gallant and Tauchen, 1996 and also Genton and de Luna, 2000, Genton and Ronchetti, 2003) to remove the bias. It can be computed in a simple iterative fashion. In this paper, we actually focus on indirect inference for bias correction in general. We rely heavily on the findings of Moustaki and Victoria-Feser (2006).

We present new M-estimators of the mean and variance of real valued random variables, based on PAC-Bayes bounds. We analyze the non-asymptotic minimax properties of the deviations of those estimators for sample distributions having either a bounded variance or a bounded variance and a bounded kurtosis. Under those weak hypotheses, allowing for heavy-tailed distributions, we show that the worst case deviations of the empirical mean are suboptimal. We prove indeed that for any confidence level, there is some M-estimator whose deviations are of the same order as the deviations of the empirical mean of a Gaussian statistical sample, even when the statistical sample is instead heavy-tailed. Experiments reveal that these new estimators perform even better than predicted by our bounds, showing deviation quantile functions uniformly lower at all probability levels than the empirical mean for non-Gaussian sample distributions as simple as the mixture of two Gaussian measures. [ABSTRACT FROM AUTHOR]

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein–Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et al. (Stat Methods Appl 19:333–354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a family of robust estimates for isotonic regression: isotonic M-estimators. We show that their asymptotic distribution is, up to an scalar factor, the same as that of Brunk''s classical isotonic estimator. We also derive the influence function and the breakdown point of these estimates. Finally we perform a Monte Carlo study that shows that the proposed family includes estimators that are simultaneously highly efficient under Gaussian errors and highly robust when the error distribution has heavy tails. [Copyright &y& Elsevier]

ABSTRACT In this paper, we investigate the performance of a class of M-estimators for both symmetric and asymmetric conditional heteroscedastic models in the prediction of value-at-risk. The class of estimators includes the least absolute deviation (LAD), Huber's, Cauchy and B-estimator, as well as the well-known quasi maximum likelihood estimator (QMLE). We use a wide range of summary statistics to compare both the in-sample and out-of-sample VaR estimates of three well-known stock indices. Our empirical study suggests that in general Cauchy, Huber and B-estimator have better performance in predicting one-step-ahead VaR than the commonly used QMLE. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to robust estimation based on dual divergences estimators for parametric models in the framework of right censored data. We give limit laws of the proposed estimators and examine their asymptotic properties through a simulation study. [Copyright &y& Elsevier]

We discuss linear regression approaches to the estimation of law-invariant conditional risk measures. Two estimation procedures are considered and compared; one is based on residual analysis of the standard least-squares method, and the other is in the spirit of the M-estimation approach used in robust statistics. In particular, value-at-risk and average value-at-risk measures are discussed in detail. Large sample statistical inference of the estimators is derived. Furthermore, finite sample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensive Monte Carlo study. Empirical results on the real data (different financial asset classes) are also provided to illustrate the performance of the estimators. [ABSTRACT FROM AUTHOR]

In this article, we propose three M-estimators for multiple regression model when response variable is subject to double censoring. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to construct interval estimators. [ABSTRACT FROM PUBLISHER]

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale. [ABSTRACT FROM AUTHOR]

In the presence of outliers, least squares estimation is inefficient and can be biased. In the 1980's several alternatives to M-estimation were proposed as attempts to overcome the lack of resistance. Least Trimmed Squares (LTS) is a viable alternative and is presently the preferred choice of Rousseeuw and Ryan (1997, 2008). Another proposed solution was S-estimation. This method finds a line that minimizes a robust estimate of the scale of the residuals. This method is highly resistant to leverage points, and is robust to outliers in the response. However, this method was also found to be inefficient. The aim of this study is to compare S-estimator with other robust estimators and the least squares estimators and also an example is given to illustrate the efficiency of S-estimator. The data used in this example are the air pollution measures. And finally a simulation study has been presented in this study. [ABSTRACT FROM AUTHOR]

We develop infinitesimally robust statistical procedures for the general diffusion processes. We first prove the existence and uniqueness of the times-series influence function of conditionally unbiased M-estimators for ergodic and stationary diffusions, under weak conditions on the (martingale) estimating function used. We then characterize the robustness of M-estimators for diffusions and derive a class of conditionally unbiased optimal robust estimators. To compute these estimators, we propose a general algorithm, which exploits approximation methods for diffusions in the computation of the robust estimating function. Monte Carlo simulation shows a good performance of our robust estimators and an application to the robust estimation of the exchange rate dynamics within a target zone illustrates the methodology in a real-data application. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to robust hypothesis testing based on saddlepoint approximations in the framework of general parametric models. As is known, two main problems can arise when using classical tests. First, the models are approximations of reality and slight deviations from them can lead to unreliable results when using classical tests based on these models. Then, even if a model is correctly chosen, the classical tests are based on first order asymptotic theory. This can lead to inaccurate -values when the sample size is moderate or small. To overcome these problems, robust tests based on dual divergence estimators and saddlepoint approximations, with good performances in small samples, are proposed. [Copyright &y& Elsevier]

In the multivariate nonlinear regression model, parameter estimators and test statistics based on least squares and maximum likelihood methods are usually nonrobust. For this type of models, we introduce M-estimators and M-tests, which are robust to departures from normality. In addition, we study the asymptotic properties and consider a computational algorithm for these estimators. [ABSTRACT FROM AUTHOR]