Asymptotically optimal tests for a regression problem of testing hypotheses

A. F. Kushnir

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 682-700

Résumé

Let $X_{1n},\dots,,X_{N_n.n}$, be observations from a sequence of series of vector independent random variables with densities $$ w(\mathbf x,s_{in});\quad s_{in}=\biggl(\frac{\theta_1\varphi_{in}^{(1)}}{\sqrt{N_n}},\dots,\frac{\theta_k\varphi_{in}^{(k)}}{\sqrt{N_n}}\biggr), $$ where $\varphi_{in}^{(j)}$ are known numbers and $\theta=(\theta_1,\dots,\theta_k)$ is a random vector with distribution $P(\theta)$. The hypothesis "all the $\theta_j=0$" is tested. It is shown that all the results concerning methods of asymptotically optimal (a.o.) test constructing proved in [1] are valid for the regression problem under consideration. If the numbers $\varphi_m^{(j)}$ satisfy some conditions, an a.o. test may be found in the class of rank-order tests (which is a generalization of the results due to Hájek [2], [4]). A.o. tests turn out to be closely related to the best tests in the sense of Pitman's asymptotic relative efficiency notion.