arXiv subjects

Ricardo A. Maronna

Publications and source records attributed to Ricardo A. Maronna.

6 recordsLinked to original sources

arXiv2025

Estimates of MM type for the multivariate linear model

We propose a class of robust estimates for multivariate linear models. Based on the approach of MM estimation (Yohai 1987), we estimate the regression coefficients and the covariance matrix of the errors simultaneously. These estimates have both high breakdown point and high asymptotic efficiency under Gaussian errors. We prove consistency and asymptotic normality assuming errors with an elliptical distribution. We describe an iterative algorithm for the numerical calculation of these estimates. The advantages of the proposed estimates over their competitors are demonstrated through both simulated and real data.

math.ST↗

arXiv2019

Optimal robust estimators for families of distributions on the integers

Let F_{θ} 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_{θ}), 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_{θ}(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.

math.ST↗

arXiv2018

Robust principal components for irregularly spaced longitudinal data

Consider longitudinal data $x_{ij},$ with $i=1,...,n$ and $j=1,...,p_{i},$ where $x_{ij}$ is the $j-$th observation of the random function $X_{i}\left( .\right) $ observed at time $t_{j}.$ The goal of this paper is to develop a parsimonious representation of the data by a linear combination of a set of $q$ smooth functions $H_{k}\left( .\right) $ ($k=1,..,q)$ in the sense that $x_{ij}\approxμ_{j}+\sum_{k=1}^{q}β_{ki}H_{k}\left( t_{j}\right) ,$ such that it fulfills three goals: it is resistant to atypical $X_{i}$'s ('case contamination'), it is resistant to isolated gross errors at some $t_{ij}$ ('cell contamination'), and it can be applied when some of the $x_{ij}$ are missing ('irregularly spaced' ---or 'incomplete'-- data). Two approaches will be proposed for this problem. One deals with the three goals stated above, and is based on ideas similar to MM-estimation (Yohai 1987). The other is a simple and fast estimator which can be applied to complete data with case- and cellwise contamination, and is based on applying a standard robust principal components estimate and smoothing the principal directions. Experiments with real and simulated data suggest that with complete data the simple estimator outperforms its competitors, while the MM estimator is competitive for incomplete data. Keywords: Principal components, MM-estimator, longitudinal .data, B-splines, incomplete data.

stat.ME↗

arXiv2015

Robust and efficient estimation of high dimensional scatter and location

We deal with the equivariant estimation of scatter and location for p-dimensional data, giving emphasis to scatter. It it important that the estimators possess both a high efficiency for normal data and a high resistance to outliers, that is, a low bias under contamination. The most frequently employed estimators are not quite satisfactory in this respect. The Minimum Volume Ellipsoid (MVE) and Minimum Covariance Determinant (MCD) estimators are known to have a very low efficiency. S-Estimators (Davies 1987) with a monotonic weight function like the bisquare behave satisfactorily for "small" p, say p not larger than 10. Rocke (1996) showed that their efficiency tends to one with increasing p. Unfortunately, this advantage is paid with a serious loss of robustness for large p. We consider three families of estimators with controllable efficiencies: non-monotonic S-estimators (Rocke 1996), MM-estimators (Tatsuoka and Tyler 2000) and tau-estimators (Lopuhaa 1991), whose performance for large p has not been explored to date. Two types of starting estimators are employed: the MVE computed through subsampling, and a semi-deterministic procedure proposed by Peña and Prieto (2007) for outlier detection. A simulation study shows that the Rocke and MM estimators starting from the Peña-Prieto estimator and with an adequate tuning, can simultaneously attain high efficiency and high robustness.

math.ST↗

arXiv2012

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 $ϕ$-mixing.

math.ST↗

arXiv2012

Consistency of M estimates for separable nonlinear regression models

Consider a nonlinear regression model : y_{i}=g(x_{i},θ)+e_{i}, i=1,...,n, where the x_{i} are random predictors x_{i} and θ is the unknown parameter vector ranging in a set Θ\subsetR^{p}. All known results on the consistency of the least squares estimator and in general of M estimators assume that either Θ is compact or g is bounded, which excludes frequently employed models such as the Michaelis-Menten, logistic growth and exponential decay models. In this article we deal with the so-called separable models, where p=p_{1}+p_{2}, θ=(α,β) with α\inA\subsetR^{p_{1}}, β\inB\subsetR^{p_{2},}and g has the form g(x,θ)=β^{T}h(x,α) where h is a function with values in R^{p_{2}}. We prove the strong consistency of M estimators under very general assumptions, assuming that h is a bounded function of α, which includes the three models mentioned above. Key words and phrases: Nonlinear regression, separable models, consistency, robust estimation.

math.ST↗