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We introduce a family of convex (concave) functions called sup (inf) of powers, which are used as generator functions for a special type of quasi-arithmetic means. Using these means, we generalize the large deviation result on self-normalized statistics that was obtained in the homogeneous case by [Q.-M. Shao, Self-normalized large deviations. Ann. Probab. 25 (1997) 285–328]. Furthermore, in the homogenous case, we derive the Bahadur exact slope for tests using self-normalized statistics.
We consider the empirical risk function (for iid ’s) under the assumption that f(α,z) is convex with respect to α. Asymptotics of the minimum of is investigated. Tests for linear hypotheses are derived. Our results generalize some of those concerning LAD estimators and related tests.
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