Bounds on asymptotic relative efficiencies of robust estimates of locations for contaminations by scale mixtures
The paper deals with robust estimation of a single location parameter \(\theta\) from a sample \(X_ 1, X_ 2,\) \(\dots, X_ n\) with \(X_ i= \theta+ Z_ i\). The errors \(Z_ i\) are independent and identically distributed, each with a cumulative distribution function \(F\) belonging to the generalized Tukey model: \[ {\mathcal F}= \bigl\{F:\;F(x)=\int [(1- \varepsilon_ 0) H(x)+ \varepsilon_ 0 H(x/ \sqrt{s})] dG(s)\bigr\}; \] \(G[s_ 1,\infty)=1\), \(\int s dG(s)= s_ 0\), with \(\varepsilon_ 0\), \(s_ 0\), \(s_ 1\) fixed, \(H\) being symmetric and absolutely continuous. The objective is to find the infimum and the supremum of the asymptotic relative efficiency of three well-known robust estimators: the Hodges - Lehmann, the sample median and the trimmed mean, with respect to the sample mean. It is interesting that these infimums and supremums remain bounded and in fact, very often are quite close to each other. As a consequence, the author is also in a position to answer the general question: given \(s_ 0\), how large is a contamination \(\varepsilon_ 0\) required for the infimum of the efficiency over \({\mathcal F}\) to exceed 1.
scientific article; zbMATH DE number 638282