Robust estimation in simultaneous equations models

In this paper we review existing work on robust estimation for simultaneous equations models. Then we discuss three strategies for obtaining estimators with a high breakdown point, a controllable efficiency, and a reasonable computational cost: (a) robustifying Three-Stages Least Squares, (b) robustifying the Full Information Maximum Likelihood method by minimizing the determinant of a robust covariance matrix of residuals, and (c) generalizing multivariate tauestimators (Lopuhaa 1991) to these models. The latter seems the most promising approach

Recent Results on Robust Estimation in Multivariate Analysis

Classic methods in multivariate analysis require the estimat.ion of mean vectors and covariance matrices, and their results can therefore be substantially altered by a small proportion of atypical observations ( "outliers")

Continuity and differentiability of regression M functionals

This paper deals with the Fisher-consistency, weak continuity and differentiability of estimating functionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes S and MM estimates. A restricted type of differentiability, called weak differentiability, is defined, which suffices to prove the asymptotic normality of estimates based on the functionals. This approach allows to prove the consistency, asymptotic normality and qualitative robustness of M estimates under more general conditions than those required in standard approaches. In particular, we prove that regression MM-estimates are asymptotically normal when the observations are $\phi$-mixing.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ368 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Optimal robust estimators for families of distributions on the integers

Let F_{{\theta}} be a family of distributions with support on the set of nonnegative integers Z_0. In this paper we derive the M-estimators with smallest gross error sensitivity (GES). We start by defining the uniform median of a distribution F with support on Z_0 (umed(F)) as the median of x+u, where x and u are independent variables with distributions F and uniform in [-0.5,0.5] respectively. Under some general conditions we prove that the estimator with smallest GES satisfies umed(F_{n})=umed(F_{{\theta}}), where F_{n} is the empirical distribution. The asymptotic distribution of these estimators is found. This distribution is normal except when there is a positive integer k so that F_{{\theta}}(k)=0.5. In this last case, the asymptotic distribution behaves as normal at each side of 0, but with different variances. A simulation Monte Carlo study compares, for the Poisson distribution, the efficiency and robustness for finite sample sizes of this estimator with those of other robust estimators.Comment: 13 page

A statistical data-based approach to instability detection and wear prediction in radial turning processes

Radial turning forces for tool-life improvements are studied, with the emphasis on predictive rather than preventive maintenance. A tool for wear prediction in various experimental settings of instability is proposed through the application of two statistical approaches to process data on tool-wear during turning processes: three sigma edit rule analysis and Principal Component Analysis (PCA). A Linear Mixed Model (LMM) is applied for wear prediction. These statistical approaches to instability detection generate results of acceptable accuracy for delivering expert opinion. They may be used for on-line monitoring to improve the processing of different materials. The LMM predicted significant differences for tool wear when turning different alloys and with different lubrication systems. It also predicted the degree to which the turning process could be extended while conserving stability. Finally, it should be mentioned that tool force in contact with the material was not considered to be an important input variable for the model.The work was performed as a part of the HIMMOVAL (Grant Agreement Number: 620134) project within the CLEAN-SKY program, linked to the SAGE2 project for geared open-rotor development and the delivery of the demonstrator part. Funding through grant IT900-16 is also acknowledged from the Basque Government Department of Education, Universities and Research

Phylogeny and chromosomal diversification in the <i>Dichroplus elongatus</i> species group (Orthoptera, Melanoplinae)

In an attempt to track the chromosomal differentiation in the Dichroplus elongatus species group, we analyzed the karyotypes of four species with classical cytogenetic and mapping several multigene families through fluorescent in situ hybridization (FISH). We improved the taxon sampling of the D. elongatus species group adding new molecular data to infer the phylogeny of the genus and reconstruct the karyotype evolution. Our molecular analyses recovered a fully resolved tree with no evidence for the monophyly of Dichroplus. However, we recovered several stable clades within the genus, including the D. elongatus species group, under the different strategies of tree analyses (Maximum Parsimony and Maximum Likelihood). The chromosomal data revealed minor variation in the D. elongatus species group's karyotypes caused by chromosome rearrangements compared to the phylogenetically related D. maculipennis species group. The karyotypes of D. intermedius and D. exilis described herein showed the standard characteristics found in most Dichroplini, 2n = 23/24, X0♂ XX♀, Fundamental number (FN) = 23/24. However, we noticed two established pericentric inversions in D. intermedius karyotype, raising the FN to 27♂/28♀. A strong variation in the heterochromatic blocks distribution was evidenced at interespecific level. The multigene families' mapping revealed significant variation, mainly in rDNA clusters. These variations are probably caused by micro chromosomal changes, such as movement of transposable elements (TEs) and ectopic recombination. These observations suggest a high genomic dynamism for these repetitive DNA sequences in related species. The reconstruction of the chromosome character "variation in the FN" posits the FN = 23/24 as the ancestral state, and it is hypothesized that variations due to pericentric inversions has arisen independently three times in the evolutionary history of Dichroplus. One of these independent events occurred in the D. elongatus species group, where D. intermedius is the unique case with the highest FN described in the tribe Dichroplini.Centro de Estudios Parasitológicos y de VectoresFacultad de Ciencias Naturales y Muse

lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers

We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the lp-averaged distances of data points from d-dimensional subspaces, where p>0. Unlike other lp minimization problems, this minimization is non-convex for all p>0 and thus requires different methods for its analysis. We show that if 0<p<=1, then for any fraction of outliers the most significant subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces the most significant subspace can be nearly recovered by lp minimization for any 0<p<=1 with an error proportional to the noise level. On the other hand, if p>1 and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.Comment: This is a revised version of the part of 1002.1994 that deals with single subspace recovery. V3: Improved estimates (in particular for Lemma 3.1 and for estimates relying on it), asymptotic dependence of probabilities and constants on D and d and further clarifications; for simplicity it assumes uniform distributions on spheres. V4: minor revision for the published versio