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This paper focuses on robust estimation in the structural errors-in-variables (EV) model. A new class of robust estimators, called weighted orthogonal regression estimators, is introduced. Robust estimators of the parameters of the EV model are simply derived from robust estimators of multivariate location and scatter such as the M-estimators, the S-estimators and the MCD estimator. The influence functions of the proposed estimators are calculated and shown to be bounded. Moreover, we derive the asymptotic distributions of the estimators and illustrate the results on simulated examples and on a real-data set.

This paper considers the multiple linear regression modelYi=xi′β+εi,i=i, …, n, wherexi's are knownp×1 vectors,βis ap×1 vector of parameters, andε1,ε2, … are stationary, strongly mixing random variables. Letβndenote anM-estimator ofβcorresponding to some score functionψ. Under some conditions onψ,xi's andεi's, a two-term Edgeworth expansion for Studentized multivariateM-estimator is proved. Furthermore, it is shown that the moving block bootstrap is second-order correct for some suitable bootstrap analog of Studentizedβn.

The asymptotic distribution of multivariate M-estimates is studied. It is shown that, in general, consistency leads to asymptotic normality and a Law of the Iterated Logarithm. The results are used to compute via matrix derivatives the asymptotic distribution of a class of estimates due to Maronna.

Let Θ be a real random variable with a known (a priori) distribution. Let the observations given Θ be iid with a common distributionF0(·)=F(·−0)F is known to belong to some family of distributions on the real line. We find estimators of Θ which are asymptotically minimax for two types of loss functions.

In the linear model Xn × 1 = Cn × pθp × 1 + En × 1, Huber's theory of robust estimation of the regression vector θp × 1 is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector En × 1. In the first model considered, the restriction of F to a set [−a0, b0] is a standard normal distribution contaminated, with probability e, by an unknown distribution symmetric about 0. In the second model, the restriction of F to [−a0, b0] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [−a0, b0] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θp × 1 are constructed, under mild regularity conditions on the sequence of design matrices {Cn × p}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets.

where thef,(B) are known functions defined on a parameter space 0 E Rk, k 2 1, and the E, are independent but not necessarily identically distributed random variables with mean values 0 and known variances. Throughout what follows 8 is assumed to be a measurable subset of Rk and P,, denotes the probability distribution of the sample x” when 19 is the parameter. Recently Sieders and Dzhaparidge [7] (abbreviated henceforth as S-D) considered the LS estimator 8, defined by

In the paper we prove rates of strong convergence of M-estimators for the parameters in a general nonlinear autoregressive model. In the proofs we utilize a variational principle from stochastic optimization theory which was proved by Shapiro (Ann. Oper. Res. 30 (1991) 169). The application of the general theory is illustrated in the case of continuous threshold models.

The Hough transform is a common computer vision algorithm used to detect shapes in a noisy image. Originally the Hough transform was proposed as a technique for detection of straight lines in images. In this paper we study the statistical properties of the Hough transform estimator in the presence of measurement errors. We consider the simple case of detection of one line parameterized in polar coordinates. We show that the estimator is consistent, and possesses a rate of convergence of the cube-root type. We derive its limiting distribution, and study its robustness properties. Numerical results are discussed as well. In particular, based on extensive experiments, we define a “rule of thumb” for the determination of the optimal width parameter of the template used in the algorithm.

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 p-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.