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 $Q_n\left(\alpha \right)=\frac{1}{n}{\sum }_{i=1}^n·f(\alpha ,Z_i)$ (for iid $Z_i$’s) under the assumption that f(α,z) is convex with respect to α. Asymptotics of the minimum of $Q_n\left(\alpha \right)$ is investigated. Tests for linear hypotheses are derived. Our results generalize some of those concerning LAD estimators and related tests.