On Huber's contaminated model |
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| Institution: | 1. School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China;2. NCMIS, KLSC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;1. Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta T6G 2G1, Canada;2. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada;3. Department of Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK;4. University of South Carolina, 1523 Greene St., Columbia SC, 29208, USA;5. Moscow Center for Fundamental and Applied Mathematics, Russian Federation;6. Steklov Institute of Mathematics, Russian Federation;7. Lomonosov Moscow State University, Russian Federation;8. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain;9. ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain;10. Universitat Autònoma de Barcelona, Spain;1. KAIST, School of Computing, Daejeon, Republic of Korea;2. LIX, CNRS, École Polytechnique, Institute Polytechnique de Paris, France |
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| Abstract: | Huber's contaminated model is a basic model for data with outliers. This paper aims at addressing several fundamental problems about this model. We first study its identifiability properties. Several theorems are presented to determine whether the model is identifiable for various situations. Based on these results, we discuss the problem of estimating the parameters with observations drawn from Huber's contaminated model. A definition of estimation consistency is introduced to handle the general case where the model may be unidentifiable. This consistency is a strong robustness property. After showing that existing estimators cannot be consistent in this sense, we propose a new estimator that possesses the consistency property under mild conditions. Its adaptive version, which can simultaneously possess this consistency property and optimal asymptotic efficiency, is also provided. Numerical examples show that our estimators have better overall performance than existing estimators no matter how many outliers in the data. |
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| Keywords: | Adaptive estimation Identifiability Minimum distance estimation Robust estimation Subsample selection |
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