Stable Asymptotics for M‐estimators. (13th April 2015)

• Record Type: Journal Article
• Authors: La Vecchia, Davide
• Is Part Of: International statistical review. Volume 84:Number 2(2016:Aug.)
• Page Start: 267
• Page End: 290
• Publication Date: 2015-04-13
• Subjects: Edgeworth expansion -- empirical likelihood -- higher‐order -- infinitesimal robustness -- p‐value -- redescending M‐estimator -- relative error -- saddlepoint techniques -- von Mises expansion; Statistics -- Periodicals; Statistics -- Bibliography -- Periodicals; Statistics -- Bibliography; Statistics -- Periodicals; Statistique; Statistique -- Périodiques; Statistique -- Bibliographie -- Périodiques; Statistique -- Étude et enseignement -- Périodiques; Statistique -- Étude et enseignement -- Bibliographie -- Périodiques; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519.2
• DOI: 10.1111/insr.12102
• Languages: English
• ISSNs: 0306-7734

Abstract

Summary: We review some first‐order and higher‐order asymptotic techniques for M ‐estimators, and we study their stability in the presence of data contaminations. We show that the estimating function ( ψ ) and its derivative with respect to the parameter ( ∇ θ ⊤ ψ ) play a central role. We discuss in detail the first‐order Gaussian density approximation, saddlepoint density approximation, saddlepoint test, tail area approximation via the Lugannani–Rice formula and empirical saddlepoint density approximation (a technique related to the empirical likelihood method). For all these asymptotics, we show that a bounded ψ (in the Euclidean norm) and a bounded ∇ θ ⊤ ψ (e.g. in the Frobenius norm) yield stable inference in the presence of data contamination. We motivate and illustrate our findings by theoretical and numerical examples about the benchmark case of one‐dimensional location model.