On the properties of compromise M-estimators optimizing weighted L2-norm of the influence function

• Published in: Nauchno-tekhnicheskiĭ vestnik informat͡s︡ionnykh tekhnologiĭ, mekhaniki i optiki Vol. 25; no. 3; pp. 527 - 535
• Main Authors: Lisitsin, D. V., Gavrilov, K. V.
• Format: Journal Article
• Language: English
• Published: ITMO University 03.07.2025
• Subjects: м-оценки; отравление данных; робастная статистика; седловая точка; сниженные оценки; устойчивые оценки; функция влияния
• ISSN: 2226-1494; 2500-0373; 2500-0373
• DOI: 10.17586/2226-1494-2025-25-3-527-535

The paper develops a theory of M -estimators optimizing the weighted L 2 -norm of the influence function. The specified criterion of the estimation quality is quite general and, in addition, allows obtaining solutions related to the class of redescending estimators, i.e., possessing the property of stability to asymmetric contamination. Such estimators, in particular, were studied within the framework of the locally stable approach of A.M. Shurygin, based on the analysis of the estimator instability functional ( L 2 -norm of the influence function), or his approach based on the model of a series of samples with random point contamination (point Bayesian contamination model). In this paper, a compromise family of estimators is studied for which the optimized functional is a convex linear combination of two basic criteria. The compromise family is similar to the conditionally optimal family of estimators proposed by A.M. Shurygin, but the criteria used can be squares of the weighted L 2 -norms of the influence function with arbitrary pre-known weight functions. The considered subject area has remained little-studied to date. In the course of the research, we used a theory we had developed earlier, which describes the properties of estimators that optimize the weighted L 2 -norm of the influence function. As a result of the study, a number of properties of compromise estimators were obtained, and the uniqueness of the family elements was shown. A family member that delivers equal values to the criteria was considered separately: it was shown that this estimator corresponds to the saddle point of the optimized functional, and is also a minimax solution with respect to the basic criteria on the set of all regular score functions. The constructed theory is illustrated using the example of the problem of mathematical expectation estimating of a normal distribution under conditions of targeted malicious influence on a data set (similar to a data poisoning attack in malicious machine learning).