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A characterization of the neighborhoods defined by certain special capacities and their applications to bias-robustness of estimates. (English) Zbl 1039.62002

Summary: A new type of neighborhood (called a (\(c,\gamma\))-neighborhood) is defined by a certain special capacity. As special cases, the neighborhood includes \(\varepsilon\)-contamination, total variation and H. Rieder’s neighborhoods [Ann. Stat. 5, 909–921 (1977; Zbl 0371.62074)]. A characterization theorem of the neighborhood and a fundamental theorem of the stochastically smallest distribution of the absolute difference between two i.i.d random variables are proved. It is shown that the median has minimax-bias among all location equivariant estimates with respect to (\(c,\gamma \))-neighborhoods.
The implosion biases of five scale estimates including MAD, S and Q over (\(c,\gamma \))-neighborhoods are derived and compared. A lower bound on the maximum asymptotic bias of an estimate of \(\theta\) over (\(c,\gamma\))-neighborhoods in a general parametric family \(\{F_0\}\) is derived. The lower bound, which is an extension of X. He and D. G. Simpson’s lower bound [ibid. 21, 314–337 (1993; Zbl 0770.62023)], depends on a parametric family \(\{(F_0- W)_{\theta}\}\) of improper distributions with some measure \(W\leqslant F_0\). For location families, the accuracy of the lower bound is investigated by using the median, and the best \(W\) is proposed. Some tables and figures of the implosion bias and the lower bound are also given in the case that the parametric family is normal.

MSC:

62A01 Foundations and philosophical topics in statistics
62F35 Robustness and adaptive procedures (parametric inference)
62G35 Nonparametric robustness
62F10 Point estimation
28A12 Contents, measures, outer measures, capacities
62E10 Characterization and structure theory of statistical distributions
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References:

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