Optimal robust M-estimators using Rényi pseudodistances

Using Renyi pseudodistances, new robustness and efficiency measures are defined. On the basis of these measures, new optimal robust M-estimators for multidimensional parameters, called optimal B"R"""@a-robust M-estimators, are derived using the Hampel's infinitesimal approach. The classical optimal B"i-robust estimator is particularly obtained. It is shown that the new optimal estimators are characterized by equivariance properties: equivariance with respect to reparametrizations, as well as equivariance with respect to transformations of the data set when the model is generated by a group of transformations. The performance of these estimators is illustrated by Monte Carlo simulations in the case of the Weibull distribution, as well as on the basis of real data.

[1]  V. Yohai.  Optimal robust estimates using the Kullback–Leibler divergence , 2008 .

[2]  Kris Boudt,et al.  Robust explicit estimators of Weibull parameters , 2011 .

[3]  Aida Toma,et al.  Robust tests based on dual divergence estimators and saddlepoint approximations , 2010, J. Multivar. Anal..

[4]  I. Vajda,et al.  Decomposable Pseudodistances and Applications in Statistical Estimation , 2011, 1104.1541.

[5]  Alfio Marazzi,et al.  Optimal robust estimates using the Hellinger distance , 2010, Adv. Data Anal. Classif..

[6]  Matthias Kohl,et al.  The cost of not knowing the radius , 2008, Stat. Methods Appl..

[7]  Z. Q. Lu.  Statistical Inference Based on Divergence Measures , 2007 .

[8]  Aida Toma,et al.  Dual divergence estimators and tests: Robustness results , 2009, J. Multivar. Anal..

[9]  H. Rieder.  Robust asymptotic statistics , 1994 .

[10]  F. Hampel.  The Influence Curve and Its Role in Robust Estimation , 1974 .

[11]  Amor Keziou,et al.  New estimates and tests of independence in semiparametric copula models , 2010, Kybernetika.

[12]  Igor Vajda,et al.  Several applications of divergence criteria in continuous families , 2009, Kybernetika.

[13]  Werner A. Stahel,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[14]  Aida Toma,et al.  Optimal robust M-estimators using divergences , 2009 .

[15]  W K Fung,et al.  Method of medians for lifetime data with Weibull models. , 1999, Statistics in medicine.

[16]  D G Hoel,et al.  A representation of mortality data by competing risks. , 1972, Biometrics.

[17]  Michel Broniatowski,et al.  Parametric estimation and tests through divergences and the duality technique , 2008, J. Multivar. Anal..

[18]  F. Vonta,et al.  On Properties of the (Φ, a)-Power Divergence Family with Applications in Goodness of Fit Tests , 2012 .

[19]  Hisayuki Tsukuma.  Minimax covariance estimation using commutator subgroup of lower triangular matrices , 2014, J. Multivar. Anal..

[20]  Douglas G. Simpson,et al.  Robust Direction Estimation , 1992 .

[21]  A. Karagrigoriou,et al.  On Distributional Properties and Goodness-of-Fit Tests for Generalized Measures of Divergence , 2010 .

[22]  A. Rényi.  On Measures of Entropy and Information , 1961 .

[23]  T. Rothenberg.  Identification in Parametric Models , 1971 .

[24]  Matthias Kohl,et al.  Infinitesimally Robust estimation in general smoothly parametrized models , 2009, Stat. Methods Appl..

[25]  Byron J. T. Morgan,et al.  Detecting parameter redundancy , 1997 .