Multivariate τ-estimators for location and scatter

We discuss the robustness and asymptotic behaviour of τ-estimators for multivariate location and scatter. We show that τ-estimators correspond to multivariate M-estimators defined by a weighted average of redescending ψ-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that τ-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, τ-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted ψ-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.     

M-estimation in regression models for censored data

In this paper, we study M-estimators of regression parameters in semiparametric linear models for censored data. A class of consistent and asymptotically normal M-estimators is constructed. A resampling method is developed for the estimation of the asymptotic covariance matrix of the estimators.     

Bounded influence estimation for regression and scale

Bednarski and Müller [Optimal bounded influence regression and scale M-estimators in the context of experimental design, Statistics 35 (2001), pp. 349–369] introduced a class of bounded influence M estimates for the simultaneous estimation of regression and scale in the linear model with normal errors by solving the corresponding normal location and scale problem at each design point. This limits the proposal to regressor distributions with finite support. Based on their approach, we propose a slightly extended class of M estimates that is not restricted to finite support and is numerically easier to handle. Moreover, we employ the even more general class of asymptotically linear (AL) estimators which, in addition, is not restricted to normal errors. The superiority of AL estimates is demonstrated by numerical comparisons of the maximum asymptotic mean-squared error over infinitesimal contamination neighbourhoods.     

The influence function of penalized regression estimators

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.     

Asymptotic properties of one-step M-estimators

We study the asymptotic behavior of one-step M-estimators based on not necessarily independent identically distributed observations. In particular, we find conditions for asymptotic normality of these estimators. Asymptotic normality of one-step M-estimators is proven under a wide spectrum of constraints on the exactness of initial estimators. We discuss the question of minimal restrictions on the exactness of initial estimators. We also discuss the asymptotic behavior of the solution to an M-equation closest to the parameter under consideration. As an application, we consider some examples of one-step approximation of quasi-likelihood estimators in nonlinear regression.     

De l'unicité des estimateurs robustes en régression lorsque le paramètre d'échelle et le paramètre de la régression sont estimés simultanément

Several methods have been suggested to calculate robust M- and G-M -estimators of the regression parameter β and of the error scale parameter σ in a linear model. This paper shows that, for some data sets well known in robust statistics, the nonlinear systems of equations for the simultaneous estimation of β, with an M-estimate with a redescending ψ-function, and σ, with the residual median absolute deviation (MAD), have many solutions. This multiplicity is not caused by the possible lack of uniqueness, for redescending ψ-functions, of the solutions of the system defining β with known σ; rather, the simultaneous estimation of β and σ together creates the problem. A way to avoid these multiple solutions is to proceed in two steps. First take σ as the median absolute deviation of the residuals for a uniquely defined robust M-estimate such as Huber's Proposal 2 or the L₁-estimate. Then solve the nonlinear system for the M-estimate with σ equal to the value obtained at the first step to get the estimate of β. Analytical conditions for the uniqueness of M and G-M-estimates are also given.     

Lp-estimators in ARCH models

For ergodic ARCH processes, we introduce a one-parameter family of L_p-estimators. The construction is based on the concept of weighted M-estimators. Under weak assumptions on the error distribution, the consistency is established. The asymptotic normality is proved for the special cases p=1 and 2. To prove the asymptotic normality of the L₁-estimator, one needs the existence of a density of the squares of the errors, whereas for the L₂-estimator the existence of fourth moments is assumed. The asymptotic covariance matrix of the estimator depends on the unknown parameter which can be substituted by consistent estimators. For the L₁-estimator we construct a kernel estimator for the unknown density of the square of the errors.     

Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

Minimax Asymptotic Mean-Squared-Error of L-Estimators of Scale Parameter

We derive the AMSE (maximal asymptotic mean-squared-error) of the general class of L-estimators of scale that are location-scale equivariant and Fisher consistent. For non-normal error distributions, we determined estimators that have minimum AMSE over the subclass of (i) α-interquantile ranges and (ii) mixtures of at most two α-interquantile ranges. Finally, the L-estimators of scale symmetrized about the median were found to have the same AMSE as their nonsymmetrized counterparts, thus yielding the same results as in the symmetrized case.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.     

Estimation of a scale parameter in mixture models with unknown location

Estimation of the scale parameter in mixture models with unknown location is considered under Stein's loss. Under certain conditions, the inadmissibility of the “usual” estimator is established by exhibiting better estimators. In addition, robust improvements are found for a specified submodel of the original model. The results are applied to mixtures of normal distributions and mixtures of exponential distributions. Improved estimators of the variance of a normal distribution are shown to be robust under any scale mixture of normals having variance greater than the variance of that normal distribution. In particular, Stein's (Ann. Inst. Statist. Math. 16 (1964) 155) and Brewster's and Zidek's (Ann. Statist. 2 (1974) 21) estimators obtained under the normal model are robust under the t model, for arbitrary degrees of freedom, and under the double-exponential model. Improved estimators for the variance of a t distribution with unknown and arbitrary degrees of freedom are also given. In addition, improved estimators for the scale parameter of the multivariate Lomax distribution (which arises as a certain mixture of exponential distributions) are derived and the robustness of Zidek's (Ann. Statist. 1 (1973) 264) and Brewster's (Ann. Statist. 2 (1974) 553) estimators of the scale parameter of an exponential distribution is established under a class of modified Lomax distributions.     

Discussion of “The power of monitoring: how to make the most of a contaminated multivariate sample” by Andrea Cerioli,Marco Riani,Anthony C. Atkinson and Aldo Corbellini

This paper discusses the contribution of Cerioli et al. (Stat Methods Appl, 2018), where robust monitoring based on high breakdown point estimators is proposed for multivariate data. The results follow years of development in robust diagnostic techniques. We discuss the issues of extending data monitoring to other models with complex structure, e.g. factor analysis, mixed linear models for which S and MM-estimators exist or deviating data cells. We emphasise the importance of robust testing that is often overlooked despite robust tests being readily available once S and MM-estimators have been defined. We mention open questions like out-of-sample inference or big data issues that would benefit from monitoring.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.     

A Class of Estimators of Population Variance Using Auxiliary Information in Sample Surveys

This article advocates the problem of estimating the population variance of the study variable using information on certain known parameters of an auxiliary variable. A class of estimators for population variance using information on an auxiliary variable has been defined. In addition to many estimators, usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999), and Kadilar and Cingi's (2006) estimators are shown as members of the proposed class of estimators. Asymptotic expressions for bias and mean square error of the proposed class of estimators have been obtained. An empirical study has been carried out to judge the performance of the various estimators of population variance generated from the proposed class of estimators over usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999) and Kadilar and Cingi's (2006) estimators.     

Role of weights in improving the efficiency of Kim and Warde's mixed randomized response model

ABSTARCT

In this paper we have suggested a class of unbiased estimators of π_S, the proportion of respondents possessing a sensitive attribute A using mixed randomized response model. The variance of the proposed class of estimators has been obtained. In addition to Kim and Warde's (2005) estimator, several other acceptable estimators of π_S have been identified from the proposed class for suitable weights. It has been shown that the newly identified estimators are more efficient than the Kim and Warde's (2005) estimator. Numerical illustrations and graphs are also given in support of the present study.     

Improved robust Bayes estimators of the error variance in linear models

We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over the entire class of scale mixture of Gaussian distributions.     

Estimation of Bowley's Coefficient of Skewness in the Presence of Auxiliary Information

In this paper, we suggest regression-type estimators for estimating the Bowley's coefficient of skewness using auxiliary information. To the first degree of approximation, the bias and mean-squared error expressions of the regression-type estimators are obtained, and the regions under which these estimators are more efficient than the conventional estimator are also determined. Further, a general class of estimators of the Bowley's coefficient of skewness is defined along with its properties. A class of estimators based on estimated optimum values is also defined. It is shown to the first degree of approximations that the variance of the class of estimators based on estimated optimum values is the same as that of the minimum variance of the proposed class of estimators. A simulation study is carried out to demonstrate the performance of the proposed difference estimator over the usual estimator.     

M-Estimators Based on Inverse Probability Weighted Estimating Equations with Response Missing at Random

Asymptotic properties of M-estimators with complete data are investigated extensively. In the presence of missing data, however, the standard inference procedures for complete data cannot be applied directly. In this article, the inverse probability weighted method is applied to missing response problem to define M-estimators. The existence of M-estimators is established under very general regularity conditions. Consistency and asymptotic normality of the M-estimators are proved, respectively. An iterative algorithm is applied to calculating the M-estimators. It is shown that one step iteration suffices and the resulting one-step M-estimate has the same limit distribution as in the fully iterated M-estimators.     

Comparisons of asymptotic biases and variances of m-estimators of scale under asymmetric contamination

We study robustness properties of two types of M-estimators of scale when both location and scale parameters are unknown: (i) the scale estimator arising from simultaneous M-estimation of location and scale; and (ii) its symmetrization about the sample median. The robustness criteria considered are maximal asymptotic bias and maximal asymptotic variance when the known symmetric unimodal error distribution is subject to unknown, possibly asymmetric, £-con-tamination. Influence functions and asymptotic variance functionals are derived, and computations of asymptotic biases and variances, under the normal distribution with ε-contamination at oo, are presented for the special subclass arising from Huber's Proposal 2 and its symmetrized version. Symmetrization is seen to reduce both asymptotic bias and variance. Some complementary theoretical results are obtained, and the tradeoff between asymptotic bias and variance is discussed.