F tests that are specially powerful for selected alternatives are built for sub-normal models. In these models the observation vector is the sum of a vector that stands for what is measured with a normal error vector, both vectors being independent. The results now presented generalize the treatment given by Dias (1994) for normal fixed-effects models, and consider the testing of hypothesis on the ordering of mean values and components.

Let (X, Y) be a random couple in S×T with unknown distribution P. Let (X1, Y1), …, (Xn, Yn) be i.i.d. copies of (X, Y), Pn being their empirical distribution. Let h1, …, hN:S↦[−1, 1] be a dictionary consisting of N functions. For λ∈ℝN, denote fλ:=∑j=1Nλjhj. Let ℓ:T×ℝ↦ℝ be a given loss function, which is convex with respect to the second variable. Denote (ℓ•f)(x, y):=ℓ(y; f(x)). We study the following penalized empirical risk minimization problem $${\widehat{\lambda }}^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P_n(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right],$$ which is an empirical version of the problem $${\lambda }^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right]$$ (hereɛ≥0...

Generalizations of the additive hazards model are considered. Estimates of the regression parameters and baseline function are proposed, when covariates are random. The asymptotic properties of estimators are considered.